A MODEL to PREDICT POCKETING POWER LOSSES in SPIRAL BEVEL and HYPOID GEARS THESIS Presented in Partial Fulfillment of the Requi

A MODEL TO PREDICT POCKETING POWER LOSSES IN SPIRAL BEVEL AND HYPOID GEARS

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Graduate School of The Ohio State University

By Erdem ERKILIÇ, B.S. Graduate Program in Mechanical Engineering

The Ohio State University 2012

Thesis Committee:

Dr. Ahmet Kahraman, Advisor

Dr. Brian Harper

ABSTRACT

In this study, a computational methodology is proposed for prediction of power losses due to pocketing (pumping or squeezing) of oil at the mesh interfaces of spiral bevel and hypoid gears. The model employs an existing manufacturing cutting simulation procedure to define surface geometries of spiral bevel and hypoid gears cut through face-milling or face-hobbing processes. With the tooth surfaces defined, a novel hypoidal discretization method is proposed to define pockets of volume between the gear teeth in mesh along the face width and circumferential directions. With the volumes of each discrete pocket along with the exit areas and associated centroids as inputs, an existing fluid mechanics formulation that utilizes principles of conservation of mass, conservation of momentum and conservation of energy is used to compute load-independent power losses due to fluid pocketing. In the end, results of various simulations representative of typical automotive and aerospace conditions are presented to quantify pocketing losses within the operating speed and parameter ranges.

ii DEDICATION

Dedicated to my family; my mom, my dad and my grandparents, who have spent

their entire lives to support me and raise me up to become the person I am today.

iii ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Prof. Dr. Ahmet Kahraman,

for this great research opportunity. I am more than thankful for his motivational support at all

stages of my studies and his endless efforts in reviewing this thesis despite his extremely busy

schedule. The inspiration and guidance he provided throughout my graduate studies have

certainly made my academic life at The Ohio State University a remarkable one. I would also like

to thank Dr. Brian Harper for accepting to be a part of my Masters examination committee.

In addition, I would like to thank all the sponsor of the Gear and Power Transmission

Research Laboratory, especially General Motors for their continuous trust and financial support

throughout my research studies. I would like to extend my gratitude to former GearLab students

and brilliant gear engineers, Dr. Mohsen Kolivand, Dr. Mohammad Hotait and Dr. David Talbot

for sharing their technical knowledge and expertise to help me advance in my studies. Also, I

would like to thank Dr. David Talbot for the enlightening discussions regarding my research and

his patient efforts in pre-reviewing this thesis. Last, but not the least, I would like to thank all of

my lab mates for their invaluable friendship during my time at the GearLab.

Finally, I would like to express my deepest love, appreciation and gratitude to the love of

my life, who stood and walked beside me in every circumstance and to my parents and

grandparents for their never-ending support at every moment and aspect of my life. Although I

was miles away, I felt their warmth and strength with me all the time throughout this journey.

Without them, I would certainly not be the person I am today. Thank you, for everything.

iv VITA

May 12, 1986 ...... Born – Adana, Turkey

June 2004 ...... Tarsus American College Tarsus, Turkey

August 2007 – May 2008 ...... Exchange, Mechanical Engineering University of California, Berkeley Berkeley, CA

January 2010 ...... B.S. Mechanical Engineering Middle East Technical University Ankara, Turkey

August 2010 – March 2010 ...... Design Engineer Arçelik Applicances Ankara, Turkey

September 2010 – Present ...... Graduate Research Associate The Ohio State University Columbus, OH

FIELDS OF STUDY

Major Field: Mechanical Engineering

Focus of Hypoid Gears, Load-Independent Power Loss due to Pocketing

v TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... iii

Acknowledgements ...... iv

Vita ...... v

List of Tables ...... viii

List of Figures ...... ix

Nomenclature ...... xi

1. INTRODUCTION ...... 1

1.1. Background and Motivation ...... 1

1.2. Literature Review ...... 5

1.3. Scope and Objectives ...... 11

1.4. Thesis Outline ...... 13

2. POCKETING POWER LOSS COMPUTATION METHODOLOGY ...... 15

2.1. Introduction ...... 15

2.2. Model Formulation and Solution Procedure ...... 18

2.2.1. Generation of Surfaces ...... 18

vi 2.2.2. Discretization of Geometry ...... 25

2.2.3. Identification of Pockets ...... 29

2.2.4. Fluid Mechanics Solution ...... 38

2.3. Summary ...... 51

3. SIMULATIONS AND PARAMETRIC STUDIES ...... 53

3.1. Introductio ...... 53

3.2. Parametric Studies ...... 54

3.2.1. Low-speed Simulations with High Oil-to-air Ratios ...... 58

3.2.1.1. Influence of Shaft Offset ...... 58

3.2.1.2. Influence of Shaft Misalignments ...... 60

3.2.2. High-speed Simulations with Low Oil-to-air Ratios ...... 67

3.2.2.1. Influence of Oil-to-Air Ratio ...... 67

3.3. Results and Discussions...... 71

4. CONCLUSION ...... 73

4.1. Summary and Conclusions ...... 73

4.2. Thesis Contributions ...... 74

4.3. Recommendations for Future Work ...... 75

References ...... 77

vii LIST OF TABLES

Table Page

2.1. Machine setting parameters ...... 19

2.2. Cutter parameters ...... 23

3.1. Basic design paremeters of the face-hobbed hypoid gear set ...... 55

3.2. Basic Design Parameters of the face-milled spiral bevel gear set ...... 56

3.3. Simulation matrix for the parametric studies ...... 57

3.4. Proportionality of pocketing power loss, , to pinion speed, Ω, for Gear Set C at different values of oil-to-air ratios, ξ0.01, ξ0.03, and ξ0.05 ...... 70

viii LIST OF FIGURES

Figure Page

1.1. Various types of cross axis gears based on shaft arrangements ...... 2

2.1. Flowchart of the overall computation methodology ...... 17

2.2. Traditional cradle-based hypoid generator ...... 19

2.3. Two main hypoid gear cutting processes: (a) face-milling and (b) face-hobbing ...... 21

2.4. Cutter geometry, (a) cutter head, (b) blade, and (c) cutting edge ...... 23

2.5. Hypoidal discretization method ...... 26

2.6. Section profiles: (a) gear section profile and (b) pinion section profile...... 28

2.7. Sectional view of pocket variation through a mesh cycle highlighting the changes to an arbitrary pocket A ...... 30

2.8. Three dimensional sketch of a pocket along the face width direction ...... 31

2.9. Three dimensional sketch of a discretized control volume ...... 33

2.10. Illustration of calculation of the normal exit area with a coarse mesh ...... 36

2.11. Variation of (a) pocket volume, (b) normal exit area at the toe side, (c) normal exit area at the heel side, (d) radial exit area at the backlash side, and (e) radial exit area at the contact side through a mesh cycle ...... 39-41

2.12. A simplified example of pocket matrix at any time increment ...... 43

ix 2.13. Variation of (a) pocket pressure, (b) normal exit velocity at the toe side area, (c) normal exit velocity at the heel side area, (d) radial exit velocity at the backlash area, and (e) radial exit velocity at the contact side area through a mesh cycle ...... 47-49

3.1. Variations in pocketing power loss, , with pinion speed, Ω, for face-hobbed hypoid gear paris with shaft offsets, 15 and 30 , and no shaft misalignments operating under simulated dip lubrication conditions at oil-to-air ratio ξ0.80 ...... 59

3.2. Definition of shaft misalignments ...... 61

3.3. Variations in pocketing power loss, , with pinion speed, Ω, for face-hobbed hypoid gear paris with shaft offsets, 15 and different values of

shaft misalignments (a) 0.2 , (b) 0.2 , and (c) 0.2 operating under simulated dip lubrication conditionsat oil-to-air ratio ξ0.80 ...... 63-64

3.4. Variations in pocketing power loss, , with pinion speed, Ω, for face-hobbed hypoid gear paris with shaft offsets, 30 and different values of

shaft misalignments (a) 0.2 , (b) 0.2 , and (c) 0.2 operating under simulated dip lubrication conditionsat oil-to-air ratio ξ0.80 ...... 65-66

3.5. Variations in pocketing power loss, , with pinion speed, Ω, for a face-milled spiral bevel gear pairs with no shaft misalignments, operating under simulated jet lubrication conditions at different values of oil-to-air ratio ξ 0.01, ξ 0.03, and ξ0.05 ...... 68

x NOMENCLATURE

Symbol Description

Area

Blade angle

Distance between consecutive sections along the face width

Array representing coordinates of a centroid

Constant volume specific heat

Specific heat ratio of air

Machine root angle

Shortest distance between mating tooth surfaces

δ Shaft misalignment

Blade offset angle

Internal energy

Blank offset

Shaft offset

Cradle angle

Cutter phase angle

Blank phase angle

Number of face width divisions

Tilt angle

Number of time increments

xi Swivel angle

Number of pockets in a certain face width division and at a given time step

Machine center to back

Number of triangular meshing elements

Hook angle

Number of teeth (or blades)

Pocketing power loss

Pressure

Cradle angle change

Specific gas constant of air

Ratio of roll

Ratio of number of blades to number of teeth

Cutter radius

Position vector

Density

Radial setting

Distance along the cutting edge from the origin of the local frame

Semiperimeter of a triangle

Temperature

Rake angle

Velocity

Volume

Work

Angular Speed

xii

Ω Pinion speed

Array representing coordinates of arbitrary point

Sliding base

Volumetric oil-to-air ratio

,, Sides of a triangle

,, Coordinates of a point

Subscript Description

Ambient

Cradle

Cutter

Offset direction

Radial exit

Normal exit

Equivalent

Gear axial direction

Gear

Oil

Pinion axial direction

Pinion

Triangular mesh

Vertex

xiii

Superscript Description

Component related to cradle rotation

Component related to cutter rotation

Index Description

Face width element

Time increment

Circumferential element

Triangular mesh element

xiv

CHAPTER 1

INTRODUCTION

1.1 Background and Motivation

Gears are traditionally specified according to their shaft arrangements. In many applications, highly efficient parallel-axis gears such as spur and helical gears must be replaced

by cross-axis gears such as hypoid and spiral bevel gear to transmit power between non-parallel

(perpendicular) shafts. Spiral bevel and hypoid gears are particularly common in many power train systems. Spiral bevel gears have intersecting gear axes while hypoid gears have a shaft offset between the two axes (i.e. they do not intersect). Many different industrial applications ranging from automotive (front and rear axles and power take-off units) to aerospace (helicopter gearboxes) use spiral bevel or hypoid gears.

Hypoid gears constitute the most general case of gear geometry as other types of gears can be obtained by simply setting certain gear parameters accordingly [1]. A hypoid gear pair with no shaft offset is reduced to a spiral bevel gear pair. A right-angle gear pair with a shaft offset that is equal to the radius of the gear is essentially a worm gear as shown in Figure 1.1.

Going from a worm gear to spiral bevel gear, the efficiency increases while the sliding velocities

1 decrease along with the pinion size. Thus, with the increased pinion size due to the shaft offset, hypoid gear pairs can have higher contact ratio and fatigue strength compared to spiral bevel gears [2].

Figure 1.1. Various types of gears based on shaft arrangements [2]

Two main cutting processes, Face-Milling (FM) and Face-Hobbing (FH), are used in the manufacturing of spiral bevel and hypoid gears. Single indexing is used in the FM process (i.e. teeth are machined one at a time) whereas the FH process uses continuous indexing resulting in accelerated manufacturing, which makes it suitable for high-volume production purposes as in the automotive industry [3].

Several factors are taken into account during the design phase of spiral bevel and hypoid gears with gear set efficiency being one of the most important factors. Increasing fuel prices and competition brought a high demand for more efficient power trains and axles. Gear engineers, therefore, continuously attempt to decrease the power losses of transmissions and drivetrains.

Along with the experimental studies that quantify these losses, theoretical models are becoming increasingly critical to the efforts to understand the mechanisms behind gear pair power losses and to devise guidelines to reduce them.

The power losses within a gearbox or an axle can be grouped into two categories based on their dependence on the power transmitted: (i) load-dependent (mechanical) power losses, and

(ii) load-independent (spin) power losses. Load-dependent power losses are induced by friction due its sliding and rolling components at the gear and bearing contacts. Load-independent power losses, on the other hand, are due to viscous effects of the lubricant. They include drag losses in the form of oil churning or air windage losses depending on the type of fluid medium, and pocketing losses that take place at the gear mesh interface due to pumping of the oil or air. Each of these power loss components must be tackled with thorough investigations, both experimentally and analytically, in order to develop validated models for designing high efficiency gear systems.

Although mechanical power losses are critical at low-speed and high-load conditions, load-independent losses dominate under low-load and high-speed operations. In many cases, attempts to minimize mechanical losses in a gearbox may have unfavorable effects on spin losses, resulting in a decrease of efficiency of the overall system [4]. This balancing act requires a good understanding of the load-independent power losses. It is essential that a model to predict load- independent power losses must account for not only drag losses but also fluid pocketing losses at the gear mesh zone. These losses not only contribute a considerable percentage of the spin power losses, but also result in increased gear noise due to the fluid trapped between the gear teeth and hence, must be minimized as much as possible [4]. In applications at high speeds where load- independent power losses are more crucial, the corresponding magnitudes of fluid pocketing losses become significant as well.

Experimental methods allow measuring of load-independent power losses by running the gear pairs at unloaded conditions (i.e. zero power transmitted such that power input equals power loss). Determining contributions of the drag and pocketing components, however, is not feasible to do experimentally since the drag loss of a single rotating gear (without its meshing partner) often different from the same with meshing gear. Hence, an analytical model that accounts for all the components of load-independent power losses is critical. This study focuses mainly on investigating the fluid pocketing losses at the mesh interface of spiral bevel and hypoid gears.

Combining a previously developed fluid mechanics model with generation of gear geometry, this thesis proposes a computational procedure to predict the component of spin losses due to fluid pocketing. Parametric studies simulating different systems from automotive and aerospace applications are also performed in order to quantify the influence of factors such as shaft misalignments, shaft offset, rotational speed, and oil-air ratio on pocketing losses.

1.2 Literature Review

Earlier research on power losses of gear pairs focused primarily parallel-axis gears.

Petry-Johnson et al [5] conducted experiments on spur gears under high speeds and various torque values to measure the influence of parameters such as gear module, tooth surface roughness amplitudes and operating conditions on spin and mechanical losses of a gear pair.

Handschuh and Kilmain [6] carried on both experimental and analytical investigations on high- speed helical gear trains that have particular use in the aerospace industry. Their results showed that increasing speed has a negative effect on the efficiency.

Various approaches were employed in investigating tooth contact friction and load- dependent (mechanical) power losses of parallel-axis gears. Benedict and Kelley [7] performed experiments with cylindrical rollers to investigate the gear tooth friction. They presented their results as an empirical formula to predict the instantaneous friction coefficient. However, a very limited range of variables within which the experiments were run bound the validity of this equation. Diab et.al. [8] derived a semi-empirical traction formula based on experiments on a two-disk test rig at low rotational speeds. They compared their simulations with the experiments of Benedict and Kelley [7] to show that discrepancies occur near the pitch line. Xu et al [9] used an EHL model along with a multiple regression analysis to obtain a new friction coefficient formula which they used in predicting mechanical efficiency of parallel axis gear pairs. Li and

Kahraman [10] extended this study to a transient mixed EHL lubrication model that is capable of accounting for the time varying contact parameters. The same authors [11] used this model to predict the spur gear mechanical power losses. Moreover, Li et al [12] used this spur gear mechanical efficiency model together with a gear design optimization model to show that measures to maximize the mechanical gear efficiency often impacts the other noise and durability

5 metrics adversely. The final design must be a compromise that delivers reasonable efficiency levels with reasonably low vibration excitations and contact and bending stresses. Later, Li and

Kahraman [13] generalized the regression based method of Xu et al [9] to derive separate closed- form formulae for both rolling and sliding friction components.

The research on load-independent (spin) losses of parallel-axis gears has primarily been experimental under different modes of lubrication. Boness [14] performed experiments on a disc and a gear operating partially submerged in lubricant to measure drag torque and estimate churning losses. Based on these experiments, he obtained empirical relations for churning losses within the ranges of the experiments. Luke and Olver [15] conducted experiments on individual and meshed spur gears to measure the churning losses. Their results were used to show certain discrepancies formulae of Boness [14]. Changenet and Velex [16] also predicted churning losses in a single and a pair of gears. Their study was based on results from a dimensional analysis and compared well with the experiments they conducted for validation. Later, Seetharaman and

Kahraman [17] proposed a physics-based model of to predict spin losses of a spur gear pair including drag and pocketing loss components. They investigated the impact of static oil level, speed, module and face width on the load independent losses. Then in a companion study [18], they compared their model to spur gear experiments performed under dip lubrication conditions to show good correlation. Recently, another study by Seetharaman and Kahraman [4] replaced incompressible flow formulations in their previous work [17] with a compressible one to predict spin losses under windage conditions with air as the medium around the gears.

Dawson [19] performed experiments on large spur and helical gears to measure the windage losses and quantify the effects of speed, gear size and geometry as well as the shape of the enclosure. The same author [20] later extended his studies to smaller gears and presented an

6 empirical relation based on his experiments. Diab et al [21] investigated the windage losses of disks and gears rotating in air with no enclosures. They followed a dimensional analysis approach to derive power loss equations and conducted experiments that showed an increase in windage with face width. They also compared their experimental results with the empirical relations proposed by Dawson [20] concluding that the empirical formula overestimates the power losses. Handschuh and Hurrell [22] conducted experiments on a single gear with and without shrouds. They concluded that windage power losses could be decreased significantly by using minimum shroud clearance in both axial and radial directions. Al-Shibl et al [23] performed two-dimensional CFD analyses on a single spur gear in air to determine windage power losses. Because the CFD model was only two-dimensional, they used the correlation by

Townsend [24] to include side windage losses. They investigated the effect of a peripheral shroud on windage power losses and concluded that no optimum spacing could be reported.

Marchesse et al [25] extended the CFD approach and performed both two and three-dimensional analyses. Comparing their results with experimental results, they concluded that two-dimensional models are unsatisfactory in covering the actual flow around the teeth of the gear. Following results from their three-dimensional CFD model, they noted significant improvements when axial flanges are placed closer to the gears.

The preliminary investigations of air pumping action in a gear pair were carried on by

Houjoh et al [26]. They measured pressure and velocity of air inside the mesh area of a spur gear pair and developed a simplified model assuming the pocket space between gear teeth as a cylindrical cavity of varying cross sectional area. The experimental results indicated that dynamic behavior of air inside the pockets is governed by the gear geometry rather than speed. This

conclusion obviously lacks the fundamental effects of operational speed on load-independent

(speed-dependent) power losses of a gear pair. The authors of this study limited their

7 formulations to calculate fluid pressure and velocity only. Moreover, their model assumed the cavity at the gear side and the pinion side as a single pocket and did not account for the flow to the adjacent pocket through the backlash area. Since the circumferential connections are also closed due to tooth contact, the formulations essential did not take into account the fluid interactions between pockets along the circumferential direction. Hence, the proposed model compared well with experiments only in the compression phase of the pocketing period.

Later, Diab et al [27] developed a model for estimating the power losses due to air pocketing inside the mesh area of spur and helical gears. Their model assumed isentropic compression/expansion of air at sonic conditions, taking place within the limits defined by the intersection of pinion and gear addendum circles. They used numerical integration to calculate the axial exit areas. Unlike Houjah et al [26], they considered the fluid flow between adjacent pockets. In their formulations, they used dimensionless and discretized forms of the continuity equation and principles of thermodynamics. Thus, the power loss due to pocketing was calculated from the heat transfer between the system and surrounding fluid media, neglecting any heat loss from the fluid to the gears. On the other hand, models by Seetharaman and Kahraman [4,17] calculating spin power losses of spur gear pairs were based on implementation of fluid mechanics principles and accounted for pocketing/squeezing of the lubricant within the mesh zone. They considered that the initial mesh position of a pocket being formed at the start of active profile. In calculating the fluid discharge areas of discretized pockets, they analytically integrated a first- order Taylor series approximation of the involute profile equation. In their model, they accounted for the interactions between pockets both in circumferential and face width directions and devised a procedure based on the principles of conservation of mass and momentum, as well as the

Bernoulli’s principle to calculate pressure and velocity of pockets. Unlike Diab et al [27], they defined the power loss as the product of discharge velocity and force acting on the fluid at the

8 discharge area. The earlier model by Seetharaman and Kahraman [17], assumed an incompressible fluid with a constant density, whereas the recent extension by the same authors

[4] accounted for changes in the density of a compressible fluid.

Most recently, Talbot [28] extended the models developed for spur gears by Seetharaman and Kahraman [4,17] to helical gears. He calculated the power losses due to fluid trapping between meshing teeth as a part of his study that investigates the efficiency of planetary gear sets.

He used a numerical approach to calculate the fluid discharge areas and utilized a pocket matrix defining the interactions between the pockets and the ambient. In his formulations, Talbot [28] used principles of mass, momentum and energy to calculate the work done on the fluid by the gears, which he essentially defined as the power loss due to fluid pocketing. The most important contribution of his model to the literature was in the lubrication conditions of the gear pair. While previous models assumed either fully compressible [26,27,17] or fully incompressible [4] fluids for lubrication, Talbot [28] devised a procedure for a more realistic case of air-oil mixture. In his model for pocketing power loss, he proposed an equivalent fluid density to account for a mixture of air and oil being trapped between the gear and pinion cavities.

Research on power losses of cross-axis gears goes all the way back to Buckingham [29]

who proposed an approximation of hypoid gear efficiency by assuming that a conjugate action

between the gear teeth that was taken to equivalent to that of spiral bevel gears and the sliding

action of the pitch surfaces is equivalent to that of worm gears. He then approximated the power

loss of a hypoid gear as the sum of power losses of a spiral bevel and a worm gear. Handschuh

and Kicher [30] developed a method to analyze the thermal behavior of spiral bevel gears. They

assumed an elliptical Hertzian contact and used a simplified expression for friction coefficient as

9 a function of slide-to-roll ratio and rolling velocity. Then they employed a finite element model to determine the heat generated as a result of the relative sliding of the tooth surfaces.

In terms of hypoid gear power losses, Simon [31] used an EHL lubrication formulation along with a hypoid gear a finite-element load distribution model to predict mechanical power losses. Jia et al [32] used a multilevel-mutligrid technique to solve the implement the same EHL equations with accelerated convergence. Taking these preliminary studies on cross-axis gears one step further, Xu and Kahraman [33] developed a model to predict load dependent power losses in hypoid gear pairs. They utilized a finite element based hypoid analysis software package [34] for contact analysis and a deterministic EHL model proposed by Cioc et al [35] to predict the friction distributions. They performed experiments and validated the proposed EHL based model. They

also compared several empirical formulae for friction coefficient proposed by various authors

[36-40] with their model to show that large discrepancies occur as the sliding-to-roll ratio

approaches zero. Later, Kolivand et al [41] extended this study by utilizing the contact model

developed by Kolivand and Kahraman [1], in the process removing the dependency of the load

distribution computation on a FE package.

Investigating the components of the rear axle including cross axis gearing, experimental

studies were performed measuring overall axle power losses. Xu et al [42] performed a thermal

mapping on a vehicle rear axle and measured power losses. They identified different components

of power losses by running spin loss tests and concluded that load independent losses increase

with speed and decrease with temperature. Xu et al [43] also investigated the influence of

bearing preload and oil volume on power losses of the same rear axle unit. Identifying different

components of power losses, they concluded that an optimum bearing preload, oil volume and

operating temperature could reduce axle losses significantly.

Amongst limited studies focusing on load-independent losses is the experimental study of

Winfree [44] on a single spiral bevel gear where he used different gear baffle configurations and measured windage power losses. He identified an optimum baffle configuration and concluded that closing the inlet of the gear reduces windage losses whereas direction of rotation does not have any significant effect. Johnson et al [45] conducted experiments to investigate the effects shrouds have on windage power loss of a single spiral bevel gear and showed that optimally placed shrouds could reduce windage as much as 70%. Then, Johnson et al [46] extended this study to a shrouded spiral bevel gear pair and presented their results concluding that gear windage becomes a significant contributor to spin losses at high speeds. Rapley et al [47] applied CFD analyses to model the windage power losses of both shrouded and unshrouded configurations of a single spiral bevel gear. They also performed experiments and reported quantitative agreement, suggesting that CFD could be used to model windage losses within a certain confidence interval.

1.3 Scope and Objectives

In view of the literature review above, it is evident that bulk of the experimental and computational research on gear efficiency focused on different components of power losses in parallel-axis gears. Studies on spin losses of cross-axis gears were mostly limited to experiments.

They failed to identify the components of spin losses and did not address fluid pocketing losses either under dip or jet lubricated conditions. An analytical model of spin losses, especially the pocketing loss component, is still not available since the CFD method is not easily applicable to gear mesh problems. There is a clear need for a physics-based model to predict fluid pocketing induced power losses in cross-axis gear pairs and to estimate the power losses associated with it.

This study aims at filling this gap by investigating the losses due to fluid pocketing in spiral bevel and hypoid gears.

In accordance with this overall goal, specific objectives of this research can be listed as follows:

(i) Employ the model of Kolivand [2] and the extension by Hotait [3] to simulate the

hypoid surface generation process to define gear and pinion surfaces including the

root geometry, and discretize the hypoid surfaces.

(ii) Using the defined hypoid geometry, define volumes, exit areas and associated

centroids of the cavities between the gear and pinion teeth.

(iii) Apply of a physics-based fluid mechanics formulation of Talbot [28] developed for

calculating power losses due to fluid pocketing in helical gear pairs to discretized

hypoid geometry, and compute the load-independent power losses due to fluid

pocketing in spiral bevel and hypoid gears.

(iv) Apply the model to various automotive and aerospace systems at different ranges

of speeds and oil levels using the developed methodology.

(v) Investigate the effects of shaft offset, misalignments and oil-to-air ratio on the

power losses due to fluid pocketing.

It must be noted here that the scope of this study will be limited to spiral bevel and

hypoid gears. Although the formulations and calculations in the rest of this study will focus on

hypoid gears, all of the procedures presented in this thesis can readily be applied to spiral bevel

gears by simply setting the shaft offset to zero. A sample simulation with a spiral bevel gear set

will be presented to show the capability of the model to handle different types of cross axis gears.

A theoretical approach will be followed in the methodology with extensive computational work. Numerical methods will be employed in the solution of governing equations. Certain assumptions and simplifications will be made in order to present the proposed methodology as a design tool with reasonable computational time requirement while capturing the physical phenomena sufficiently.

The proposed model will allow the analysis of both face-milled and face-hobbed gears.

The focus of the calculations will be on the component of load independent power losses due to pocketing in the mesh interface. Simulations will be performed to represent the conditions posed by different applications with various speeds and lubrication types. Although the formulations provided in this thesis are applicable to any type of cross axis gearing, parametric studies will be limited to gear pairs at a right angle.

1.4. Thesis Outline

In Chapter 2, the computational methodology to predict the power losses due to gear mesh pocketing will be presented. The physical system will be outlined and the assumptions made in the methodology will be listed. Then, the model formulation and solution procedure will be detailed. Surface generation methods for hypoid gear pairs [2] will be summarized. The novel discretization method applied in the solution procedure will be presented along with the guidelines for identifying the cavities between meshing teeth. Finally, the fluid mechanics solution previously developed [28] for helical gears will be applied to predict power losses due to fluid trapping at the mesh area of spiral bevel and hypoid gear pairs.

Chapter 3 presents the results of simulations performed using the proposed method. Two main applications of cross axis gears, automotive rear axles and rotorcrafts, will be accounted for

13 in the simulations. For automotive applications where dip lubrications is a commonly used, power losses at low speeds and high oil-air ratios will be investigated. On the other hand, jet lubrication simulations will be made considering aerospace applications and power losses of a high-speed drive running at low oil ratios will be simulated. At the end, parametric studies will be performed to investigate the influences of shaft off-set and misalignments on the power losses in these different simulated conditions.

The last chapter of this thesis will summarize the studies presented. Major conclusions will be drawn and recommendations of extending this research will be listed as future work.

CHAPTER 2

POCKETING POWER LOSS COMPUTATION METHODOLOGY

2.1. Introduction

A computation methodology is presented in this chapter for determining the component of load independent power losses due to fluid pocketing in spiral bevel and hypoid gears. As outlined in Figure 2.1, the overall procedure uses a previously developed method for the generation of hypoid surfaces [2]. Then, a novel discretization method for spiral bevel and hypoid gears is introduced. The proposed method for discretizing the geometry reduces overall computational time requires. Having obtained the discretized hypoid geometry, procedures for identifying the cavities between the meshing teeth of the pinion and gear numerically are presented. These cavities will be referred to as “pockets” for the rest of this study.

The geometric properties of the pockets are determined. Volumes, discharge areas and associated centroids are calculated for each of the pockets. Contacts between teeth at each discrete section are determined and openings outside of the mesh area that are considered as having ambient properties are identified. As a result, a matrix of pockets with varying geometric

15 properties is obtained. This matrix extends within the mesh area in the circumferential and face width directions as well as in time.

Since, the hypoid gear pair is properly discretized and reduced to a matrix of pockets with time dependent geometric properties, it can be correlated with a parallel axis gear pair. Hence, a previously developed fluid mechanics formulation by Talbot [28] for calculating the spin power losses due to fluid pocketing in spur and helical gears is utilized. The pressure and velocity distribution throughout a complete mesh cycle is determined by considering the variation of these properties in time for all of the pockets. In the end, the component of load-independent power losses that is due to fluid trapping in the pockets is determined and averaged through a mesh cycle. Throughout the rest of this thesis, the abovementioned phenomena of fluid trapping will be shortly referred to as “pocketing” and the associated power losses averaged through the mesh cycle will be termed as “pocketing power loss”.

The identification of pockets is performed after the geometry is generated and discretized using a set of cone surfaces. It is assumed that a new pocket is formed as the pinion tooth starts to penetrate into the tooth blank of the gear. If the mating tooth surfaces are within a geometric tolerance, a contact condition is assigned numerically to account for tooth contact.

The physical system consists of a hypoid gear pair and a fluid medium. This fluid can be a certain type of lubricant where the gear pair is immersed in or a mixture of air and lubricant where a jet lubrication method is used to spray the lubricant into the mesh area. Although unlikely, the gear pair may be rotating in air only as well.

In the following sections, the surface generation process for both face-milled (FM) and face-

hobbed (FH) hypoid gear pairs is outlined. The FM process can be considered as a special

Figure 2.1. Flowchart of the overall computation methodology

case and the formulation for the surface generation can be derived from the FH process [2].

Therefore, the procedure presented here is detailed for a face-hobbed hypoid gear pair and

differences are pointed out as they emerge in the formulation for face-milled gears.

The fluid mechanics formulation developed by Talbot [28] for calculating pocketing losses for helical gears is capable of being extended to spiral bevel and hypoid gears. This physics-based model utilizes the principles of conservation of mass, momentum and energy to calculate pocketing power loss and is capable of handling compressible fluids. It accounts for an air-oil mixture with an equivalent density by using ideal gas law and assuming isentropic expansion of air. Effects due to potential energy, surface tension, electrical and magnetic properties of the medium and gears are neglected [4].

2.2. Model Formulation and Solution Procedure

The overall computation methodology outlined in Figure 2.1 is followed in this section.

Governing equations are presented and the proposed procedures are detailed. The solution procedure complies with the assumptions made in the previous section.

2.2.1. Generation of Pinion and Gear Surfaces

The complex nature of the hypoid geometry prevents closed-form representation of tooth surfaces. Instead, a set of implicit equations define the cutting process, where two gears rotate together and the teeth of one of the gears is replaced with a blade group. The method outlined in this section simulates the cutting process for hypoid surface generation. As such it requires the machine tool settings, cutter geometry and gear blank dimensions to be provided as inputs for this computation. A more detailed explanation of this procedure can be found in references [2,3].

Figure 2.2. Traditional cradle-based hypoid generator [2]

Table 2.1. Machine setting parameters

Symbol Parameter Symbol Parameter

Cradle Angle Blank Phase Angle

Radial Setting Blank Offset

Swivel Angle Machine Root Angle

Tilt Angle Machine Center to Back

Cutter Phase Angle Sliding Base

The simulation process starts from the cutter blade geometry for both face-milled (FM) and face-hobbed (FH) hypoid gears. Then, a set of matrix transformations defined through the machine settings are applied. Figure 2.2 shows an example of a traditional cradle-based hypoid generator and Table 2.1 lists the key machine setting parameters. Most of these parameters are commonly fixed while some of them are represented by a polynomial function of q, the cradle angle change, to account for higher order motions. The relative motions during the cutting process on a cradle-based hypoid generator are due to the angular speeds associated with cutter, cradle and the blank axes, and are represented by , and respectively [2,3].

The formulations presented in this section account for the two cutting processes, face- milling (FM) and face-hobbing (FH), as well as the two motion types, Formate and Generate. In

the Formate case, the surface of the gear tooth is a copy of the generating surface. On the other

hand, a roll action is applied in the Generate case, where the cradle and the gear blank rotate

synchronously. This roll action is termed as Ratio of Roll [2, 3],

2.1

where is the component of angular speed of the gear blank related to cradle rotation, is

the angular speed of the cradle.

The main difference between the FM and FH processes is the indexing method as

illustrated in Figure 2.3. Single indexing is used in the FM process, where the cutter moves

around the gear blank while it is fixed and cuts each tooth one by one. In the FH process,

continuous indexing is applied and the cutter advances around the gear blank as it rotates and

continuously cuts the teeth. In order to allow for continuous indexing in the FH process, the

cutter and the work piece rotate according to [2, 3],

Figure 2.3. Hypoid gear cutting processes: (a) face-milling and (b) face-hobbing [3]

2.2

where, is the component of angular speed of the gear blank related to cutter rotation, is

the angular speed of the cutter, is the number of blades and is the number of teeth on the gear blank.

The angular speed of the blank can be represented generically for both the FM and FH processes as [2, 3] as

2.3

where 0 for FM process due to the absence of relative rotation between the cutter and the blank as a result of single indexing. Similarly, a generalized equation that defines the blank rotation for all cases of cutting methods and motion types can be written as [2, 3],

2.4

where 0 when the motion type is Formate, due to the lack of synchronous rolling motion between the cradle and the blank. Also, 0 for the FM process because of single indexing.

The cutting edge of a typical blade consists of four sections called flankrem, profile,

toprem and edge as shown and identified in Figure 2.4. Table 2.2 lists the key parameters of the

cutter. An arbitrary point on the cutting edge has a position vector relative to the local coordinate

frame attached to the cutter [2] is given as,

2.5a

Figure 2.4. Cutter geometry, (a) cutter head, (b) blade, and (c) cutting edge [2]

Table 2.2. Cutter parameters

Symbol Parameter

Blade Angle

Rake Angle

Hook Angle

Blade Offset Angle

Cutter Radius

23 where, is the distance along the cutting edge from the origin of the local coordinate frame ,

as shown in Figure 2.4, and,

cos sin 01 , 2.5b

, , , sin 0cos 1 . 2.5c

Note that, the superscript T denotes a matrix transpose and , represents a homogeneous

4x4 transformation matrix of pure rotation with its first argument defining the rotation axis and its second argument defining the rotation angle.

The position vector of the arbitrary point A in the cutter coordinate frame is transformed to the gear coordinate system attached to the blank by applying a series of matrix transformations such that [2]

, , , , , ,

∙ , , , , . 2.6

Note that , represents a homogeneous 4x4 transformation matrix of pure rotation as explained earlier and , represents a homogeneous 4x4 transformation matrix of pure

translation, the first argument defining the translation axis and the second argument

defining the translation distance.

The transformation formulated in Eq. (2.6) results in a family of surfaces in the blank

coordinate system. The envelope of this family of surfaces is represented mathematically by the

Equation of Meshing [2]

∙ 0 2.7 which defines the generated surface on the blank.

Since the governing equations for cutting simulation and tool geometry are extremely complex, a closed-form solution for the Equation of Meshing cannot be obtained. Instead, for a predetermined domain defined by gear blank dimensions, a selected pair of the three unknown parameters , , is solved implicitly from Eq. (2.7) using a nonlinear solver [2,3].

2.2.2. Discretization of the Geometry

As explained in the previous section, closed-form relations defining the geometry of hypoid gears cannot be expressed due to extreme complexity. Therefore, at the end of the surface generation process, a set of points that lie on the tooth surfaces is obtained. For further analysis with this information, a novel discretization method is developed here.

The main goal in determining a discretization method is reducing the required CPU time to make the procedure suitable as a design tool. Since extensively iterative computations already take place in surface generation process [2] as well as the fluid mechanics calculations [28] any attempt to decrease the amount of computation at this phase of the procedure positively affects the computational efficiency of the model.

For spur and helical gears, discretization is straightforward. Sections are commonly defined through a set of transverse planes that are normal to the rotational axis of the gear teeth.

This way, as the gears incrementally rotate through the mesh cycle, the sectional geometries of each gear remain unchanged due to involute geometry and rotational axi-symmetry. Hence, a

25 simple rotation transformation can be applied to the sectional geometries of the gear and the pinion to move them to the next time increment.

Figure 2.5. Hypoidal discretization method

The challenge with spiral bevel and hypoid gears is the lack of this common plane that yields unchanged sectional geometries through a mesh cycle. Considering the unique geometry of hypoid gears defined by using a “hypoidal” coordinate frame [3], the basic cone geometry of the hypoid gear is utilized to develop a hypoidal discretization method. Hence, as illustrated in

Figure 2.5, the sections are not a set of planes, but rather a set of lateral cone surfaces. The computational burden is reduced significantly by using this discretization method, The conical section surfaces shown in Figure 2.5 extend outwards as a set of lateral cone surfaces and cover the entire face width of the gear. At each section surface, a circumferential profile of the gear and the pinion is obtained. Figure 2.6 shows an example pair of gear and pinion section profiles.

Here it is noted that the profile of the pinion section away from the mesh area is rather disrupted.

This is due to the shape of the lateral cone surface extending outwards. As the geometry of the pinion section away from the mesh is does not contribute to the pocket formation, this has no bearing on the pocketing loss computations. Section profile around the mesh area, meanwhile, is quite similar to those of spur or helical gear profiles in terms of overall shape.

The main advantage of the proposed discretization scheme becomes evident through a meshing cycle. With an arbitrarily chosen discretization method, the section profiles must be updated at each time step increment. However, the section cone used in the hypoidal discretization method is the same cone also used in the surface generation of the gear in the hypoidal coordinate system [3]. Therefore, the section profile of the gear remains unchanged throughout a mesh cycle and consecutive positions of the tooth profiles are obtained by applying a simple rotation transformation about the rotation axis of the gear, while the corresponding pinion section profiles must still be computed at each time increment. Yet, the advantage gained by applying this discretization method to determine the section profile of the gear is significant in

Figure 2.6. Section profiles: (a) gear section profile and (b) pinion section profile

28 providing sufficient computational time savings to maintain the model as a potential design tool.

An arbitrary, brute-force discretization scheme would not satisfy this objective.

2.2.3. Identification of Pockets

The discretization method introduced and applied in the previous section successfully reduces the complex problem of hypoid gear pair pocketing to an equivalent but simpler problem of helical gear pair pocketing. In order to utilize the fluid mechanics model of Talbot [28], pockets must be identified in the face width direction according to similar guidelines as in helical gears. Focusing on the mesh area within a discretized section of a hypoid gear pair, pockets are defined according to the rotation direction. As the pinion tooth moves into the cavity between the mating gear teeth, a new pocket is formed. The volume of this pocket is first squeezed and then expanded throughout the mesh cycle. Finally, as the pinion tooth leaves the mesh area, the pocket is considered to be at ambient conditions. Figure 2.7 shows a set of sectional views of the mesh area throughout a mesh cycle to illustrate this process.

The shape of a pocket in a spiral bevel or hypoid gear pair is more complex than a spur or helical gear pocket where there is a periodicity in the face width direction [28]. Due to the spiral angle of the teeth and the conical shape of the gear blank, the cross-section of a pocket changes along the face width. Depending on the circumferential position of the pocket, the cross sectional area can be maximum at the two sides of the face width and minimum at the middle as illustrated in Figure 2.8.

From a physics point of view, the power losses due to pocketing are related to the forced motion of the fluid between the control volume defined by the pockets and the ambient surroundings. Fluid medium that fills the pockets induces resistance on the rotation of the

Figure 2.7. Sectional view of pocket variation through a mesh cycle highlighting the changes to an arbitrary pocket A

Figure 2.8. Three dimensional sketch of a pocket along the face width direction

31 gears, which requires a certain amount of work to overcome. This results in a reduction of the pocket volume and an increase of pressure. Thus, the fluid trapped inside the pocket is forced to discharge from the control volume through the exit areas to the surroundings at ambient pressure.

Essentially, the gear pair acts as a compressor of a pump by pressurizing the fluid inside the pockets and pushing it towards the ambient. This action causes power loss due to pocketing.

Therefore, exit areas, volume and associated centroids must be identified and calculated numerically for every pocket. Discretizing the hypoid geometry, using a set of lateral cone surfaces, results in number of face width divisions. Also, discretizing the mesh cycle into number of time increments, the pocket identification process is repeated times.

At each face division and time increment, the points along the mating tooth surfaces of the gear and pinion with the shortest distance are searched and found methodically. These points are assigned as the starting and ending points of each pocket. Thus, the volume and exit areas are calculated for times, where represents the number of pockets in a certain face

width division and at a given time step. Note that ,, meaning the number of pockets defined at a particular face width division i ( iI[1, ] ) and time step j ([1,]jJ ) depends on

section profiles of that specific face width division at that specific time step.

For each control volume defined between two consecutive discretized pockets along the

face width direction, there are two radial exit areas, ([1,]kK ), at both sides along the ,

circumferential direction, calculated numerically by using the trapezoids formed between

consecutive sections as illustrated in Figure 2.9 [28]. Using the area formula for a trapezoid, the

radial exit areas at both sides of the control volume are given as

Figure 2.9. Three dimensional sketch of a discretized control volume

2.8 , 2 , ,

Where is the distance between consecutive sections along the face width direction, ,

and are the radial exit areas and the shortest distances between the mating tooth surfaces , on the two sides of the -th pocket in the -th face width division at the -th time step as shown

in Figure 2.9.

The radial exit areas on the drive side are referred to as contact areas and the ones on the

coast side are termed as backlash areas. The associated centroid for each of the backlash and

contact areas are calculated from a weighted average of the midpoints of the two bases of the

trapezoid such that

, , , , 2.9 , , ,

where and are the arrays that represent the centroids of the radial exit areas and , , midpoints of the bases of the trapezoids on the two sides of the -th pocket in the -th face width

division at the -th time step as shown in Figure 2.9.

For each discretized control volume, there are also two normal exit areas, , at both ,

sides along the face width direction as illustrated in Figure 2.9. Due to the rather complicated

shape of these areas, a triangular meshing algorithm is developed for numerical calculation. Each

normal exit area is divided into small triangles using the nodes along the gear and pinion

section profiles. Then, the area of each of these triangles, , is calculated using Heron’s

Formula [46],

2.10a where , , and are the three sides and the semiperimeter of the -th triangle in the mesh calculated using the vertices of the triangle such that

, 2.10b

, 2.10c

, 2.10d

2.10e 2

where , and are the coordinates of the vertices of the -th triangle in the mesh as shown in Figure 2.10.

The normal exit areas closer to the toe of the tooth are referred to as toe-side exit areas and the ones closer to the heel are termed as heel-side exit areas. The associated centroids of the toe-side and heel-side exit areas are calculated using the same triangular mesh. For each small

triangle, the centroid is determined from the average of the coordinates of the three vertices such that,

Figure 2.10. Illustration of calculation of the normal exit area with a coarse mesh

2.11 3

where represents an array with the coordinates of the vertices of the -th triangle in the

mesh.

Having calculated the areas and centroids of each small triangle within the mesh, the total

normal exit areas at both sides of the control volume are given as,

2.12 ,

and, the associated centroid of the normal exit areas are calculated using a weighted area average

of each small triangle, such that,

. 2.13 ,

Finally, the two radial exit areas along the circumferential direction and the two normal

exit areas along the face width direction that defines each discretized control volume are

identified and calculated. The volume enclosed by these exit areas is determined from the

trapezoidal prism formula as

2.14 , 2 , ,

where is the distance between consecutive sections along the face width direction and

and are the normal exit areas enclosing the control volume. , ,

Using the presented formulation, the normal and radial exit areas as well as the volume

enclosed by these areas are calculated for all of the identified pockets along the circumferential

direction. By repeating this procedure for each section along the face width direction, and at

every time step throughout a mesh cycle, a time trace of the pocket geometry can be obtained.

Following a single pocket as it moves through the mesh cycle at a given section, as shown in

Figure 2.7, the time variations of pocket exit areas and volume are obtained.

Figure 2.11 shows the results of a sample calculation. It is noted that the variation of

normal exit area is similar to the variation of volume as suggested by Eq. (2.14). The radial exit

area at the drive side, referred to as contact-side area, shows a similar decay ultimately reaching

zero indicating that contact occurs. The radial exit area at the coast side, referred to as backlash

area, also decreases, but never reaches to zero. It is noted that, immediately after the pocket

forms and moves towards the middle of the mesh, it is squeezed between the gear teeth. Then, the

pocket starts to expand as it moves away from the mesh area. This results in the evident

decreasing-increasing trend of the volume and exit areas in Figure 2.11, throughout the mesh

cycle.

2.3.4. Fluid Mechanics Solution

After the volumes and exit areas, as well as their variations in time, are calculated for all discrete pockets, the hypoid pocketing problem reduces to a fluid dynamics problem independent of geometry. Hence the discrete multi-degree-of-freedom fluid mechanics formulation developed by Talbot [28] can be utilized. Since this model is completely physics-based and isolates the pocketing phenomena in helical gears from any geometric dependency, it can be extended to calculate the pocketing losses in spiral bevel and hypoid gears as well.

Figure 2.11. Variation of (a) pocket volume, (b) normal exit area at the toe side, (c) normal exit area at the heel side, (d) radial exit area at the backlash side and, (e) radial exit area at the contact side through a mesh cycle

Figure 2.11 (continued)

The identification of pockets essentially yields a time varying, 2x2 pocket matrix as

illustrated in Figure 2.12. The pockets extend along the face width and circumferential directions.

Connections between the pockets and with the ambient surroundings are established as well as

contact conditions that hinder the interconnections between the pockets. In Figure 2.12,

connections are shown by open channels that allow fluid transport between adjacent pockets and

contact conditions are denoted by closed channels, where a cross on the channel indicates a

blockage. The dotted connections represent the pockets that are not shown in the pocket matrix

for simplicity.

Note that, the pocket matrix represented in Figure 2.12 changes in time. Number of

pockets as well as the quantities that define each pocket vary together with the connections and

contact conditions. For number of pockets along the circumferential direction of number of

face width divisions discretizing the hypoid geometry, a complete mesh cycle divided into J equal

intervals results in a total number of pockets.

The fluid mechanics formulation developed by Talbot [28] uses the principles of conservation of mass, conservation of momentum and conservation of energy to calculate pocketing power loss. This model is capable of handling both compressible and incompressible fluids and accounts for air-oil mixture with an equivalent density by using ideal gas law and assuming isentropic expansion of air. A brief summary of this formulation will be presented here along with the governing equations. The detailed analysis including the discretized forms of the equations for the numerical solution of these governing equations can be found in reference [28].

Figure 2.12. A simplified example of pocket matrix at any time increment [28]

Considering the pocket matrix shown in Figure 2.12, fluid flows between the control

volumes, defined as pockets, throughout the mesh cycle. Conservation of mass, conservation of

momentum and conservation of energy for such a control volume and its boundaries are given,

respectively, as

0 2.15a

0 2.15b

2.15c

Where is the density of the fluid, is the volume of the pocket, is the exit area, is the

exit velocity, is the pressure gradient across the exit area, and is the internal

energy of the fluid, being the constant volume specific heat, and being the temperature.

Here denotes the work done by the gears on the control volume, which is by definition equivalent to the pocketing power loss. Also, indicates integration over the control volume

and indicates integration over its exit surfaces.

The fluid density in these equations can be constant, as in the case of an incompressible

lubricant. For a more common case of a compressible air-oil mixture, the density is temperature

and pressure dependent, following from the ideal gas law, such that,

. 2.16

Assuming that the density of the oil in the mixture remains constant, acting as an incompressible

fluid, and the density of the air changes as the mixture expands, acting as a compressible fluid, an

equivalent fluid density can be defined as [28]

1 2.17 1

where is the density of the oil, is the ambient pressure, is the ambient temperature ,

is the specific gas constant of air, is the temperature of air and is volumetric oil-to-air

ratio of the fluid. Note that, the temperature of the air is obtained from the ideal gas law in Eq.

(2.17) for an isentropic process, which yields

2.18

where is the specific heat ratio of air. Thus, the formulation presented here is capable of

handling a compressible air-oil mixture as the fluid. This capability of the model allows

simulating both jet and dip-lubrication conditions.

The governing equations (2.15) are applied in a discretized form to the pocket matrix

given in Figure 2.12. The mesh cycle is divided into equal intervals and the solution of this set of

differential equations is obtained for each time step using a numerical solver utilizing a first order

predictor-corrector method. This solution scheme developed by Talbot [28] obtains initial

conditions from the solution of each previous time step and repeats the process until a steady-

state solution is found.

Applying this solution procedure, pocket pressures and exit velocities at boundaries of the control volumes are calculated for each time step through the mesh cycle. Results of a sample calculation for a discrete face width division of a pocket are shown in Figure 2.13. Changes in the pressure represent a suction or pumping effect of the fluid. The positive direction for the normal and radial exit velocities is chosen as the increasing face width and circumferential directions as shown in Figure 2.12. Thus, a positive exit velocity at the backlash indicates flow into the control volume whereas at the contact side, a negative exit velocity represents inward flow. Similarly, a positive velocity at the normal exit areas indicates fluid flowing from the toe of the tooth towards the heel. Hence, flow into the control volume from a previous section is represented with a positive velocity while flow into the control volume from the next section is denoted by a negative velocity.

The fluid flow is driven mainly by the pressure gradient between adjacent pockets.

Naturally, the fluid tends to flow from a high-pressure pocket to a low-pressure pocket.

Therefore, observing the variation of pressure in a single pocket by itself is not enough to capture the entire phenomena. The spiral angle and the conical shape of the hypoid geometry also affect the fluid flow through creating a pressure gradient across the face width of the gear.

As shown in Figure 2.8, the three-dimensional sketch of a single pocket across the face width illustrates that the control volumes are smaller in the middle sections and larger at the two end sections. Therefore, a pressure gradient occurs in the face width direction from the middle of the face width towards the toe and the heel of the tooth. Hence, the fluid flows from the middle section towards the end sections. For the sample calculation presented, the discrete pocket was chosen at a face width division beyond the middle section. Thus, the fluid flow observed is

Figure 2.13. Variation of (a) pocket pressure, (b) normal exit velocity at the toe side area, (c) normal exit velocity at the heel side area, (d) radial exit velocity at the backlash side area, and (e) radial exit velocity at the contact side area through a mesh cycle

Figure 2.13 (continued)

49 unidirectional towards the heel of the tooth as indicated by normal exit velocities of the same sign in Figure 2.13(b) and Figure 2.13(c).

In the circumferential direction, a control volume experiences a compression/expansion cycle throughout the mesh. The pocket is compressed as it moves towards the middle of the mesh and expands as it moves out, as illustrated in Figure 2.7. Since the pockets compressed in the middle of the mesh are at a higher pressure than adjacent pockets at either side, the fluid tends to flow out from the middle pocket towards the two adjacent pockets in the circumferential direction. However, physical contact hindering the flow of fluid between adjacent pockets introduces an additional complexity to the phenomena. After contact occurs, as indicated by zero exit velocity in Figure 2.13(e), the direction of flow changes at the backlash exit area as well as the sign of the radial exit velocity shown in Figure 2.13(d).

Despite the compression of pocket volume, shown in Figure 2.11(a), pocket pressure in

Figure 2.13(a decreases during the compression portion of the mesh cycle. This decrease in pressure occurs as a result of increased flow through the backlash area after contact as shown in

Figure 2.13(d). In the expansion portion of the mesh cycle, as pocket volume expands, velocity of flow through the backlash area reaches to a maximum and starts to decrease, resulting in an increase of pocket pressure. When contact ends, the contact side radial exit area opens and increases as shown in Figure 2.11(e), allowing fluid to discharge to leading pockets that are closer to the ambient surroundings and at a lower pressure as shown in Figure 2.13(e). When the pocket nears the end of the mesh cycle, flow through the backlash exit area changes direction again, due to fluid flow from pockets that are in the middle of the mesh and at higher pressures.

Although the geometry of the gear pair shapes the pressure gradient, the flow between adjacent pockets is essentially driven by the forced acceleration/deceleration of fluid as a result of

the work done by the gears on the control volumes. This work is equivalent to the pocketing

power loss by definition, and is calculated from the discrete form of the conservation of energy,

Equation (2.15c), which gives the power loss of a single face width division of an individual

pocket at a certain mesh position. Summing up these losses across the circumferential and face

width direction, then averaging over a complete mesh cycle the total pocketing power loss, , is

found as

1 . 2.19 ,

where is the total number of face width divisions, is the total number of pockets along the

circumferential direction of each face width division and is the total number of mesh positions,

as introduced earlier.

2.4. Summary

In this chapter, a new methodology to calculate pocketing power loss in spiral bevel and hypoid gears is presented. Gear and pinion surfaces are generated using a formulation, based on simulation of the manufacturing process, developed by Kolivand [2] and extended by Hotait [3] to include root geometry. A novel hypoidal discretization method is introduced and implemented. The gear geometry is divided into a set of sections along the face width direction using lateral cone surfaces and section profiles are determined. At each section, pockets are identified and corresponding parameters are calculated. This process is repeated throughout the mesh cycle at incremental time steps. In the end, a time varying pocket matrix, which is independent of the gear type, is obtained. Hence, the fluid mechanics formulation developed by

Talbot [28] for calculating pocketing power loss in helical gears can be employed. The pocket

51 pressures and fluid discharge velocities are calculated. Finally, using principles of conservation of mass, conservation of momentum and conservation of energy, pocketing power loss is calculated for each pocket and averaged through the mesh cycle to obtain the average power loss due to pocketing.

CHAPTER 3

SIMULATIONS AND PARAMETRIC STUDIES

3.1. Introduction

In this chapter, the pocketing model proposed in Chapter 2 will be used to simulate

various face-hobbed and face-milled hypoid and spiral bevel gear pairs representative of

automotive and aerospace applications. In automotive industry, hypoid gears are used

extensively in rear-axle applications. The operating speeds for these applications are typically

low, bound by the maximum speed of the vehicle, and the gears are generally dip lubricated.

Experiments show that the rotation of the gears mixes the oil with air resulting a mixture of

lubricant and with a high oil ratio to be pocketed by the gear pair [42,43]. On the other hand,

spiral bevel gears are commonly used in the aerospace industry for rotorcraft applications where

the operating speeds can be an order of magnitude higher than the speeds of automotive systems.

The gear pairs in a helicopter transmission are typically jet lubricated and the air in the system

fills up the majority of the cavities between meshing teeth, resulting in a much lower oil-to-air

ratio [49,50].

3.2. Parametric Studies

A series of parametric studies were performed based on simulations of rear axle applications at low speeds (pinion speeds up to 3000 rpm corresponding to vehicle speeds up to

80 mph) and high oil-to-air ratios, as well as rotorcraft applications at high speeds (up to 18000 rpm) and low oil ratios. Influences of the shaft offset, shaft misalignments and oil-to-air ratio in the fluid mixture on the pocketing power loss are investigated. The simulations for these parametric studies were run within representative ranges of speed and oil-to-air ratios representative of the lubrication method (dip or jet). The ambient conditions were kept the same for all cases. For lubrication, a typical axle fluid (75W90) is used in all simulations with a constant oil inlet temperature of 90.

Two sets of face-hobbed hypoid gear pairs and one face-milled spiral gear pair are used

for this parametric study. Due to the high complexity and interdependence of geometric

variables, altering certain design parameters in a hypoid gear pair results in a change in various

parts of the geometry. As such, one certain design parameter cannot be varied independent of the

rest of the parameters of a design. Instead, complete individual designs must be considered.

The example face-hobbed hypoid gear sets (Set A and Set B) used in the parametric study

presented in this section are specifically generated to be reasonably close to each other in many

attributes (e.g. same numbers of teeth on pinion and gear and the same gear pitch diameters), but

different in their shaft offset values. Basic design parameters of these two gear sets are listed in

Table 3.1. The face-milled spiral bevel gear set (Set C) used in the simulations is generated using

the surface generation formulation explained in Chapter 2, with a zero shaft offset. The basic

design parameters of this example gear set are listed in Table 3.2.

Table 3.1. Basic design parameters of the face-hobbed hypoid gear sets

Gear Set A Gear Set B

Parameter Pinion Gear Pinion Gear

Number of Teeth 12 41 12 41

Hand of Spiral Left Right Left Right

Face Width (mm) 33.1 31.2 36.8 32.3

Mean Spiral Angle (deg) 40.0 31.3 47.0 29.6

Pinion Shaft Angle (deg) 90 - 90 -

Pinion Shaft Offset (mm) 15 - 30 -

Gear Pitch Diameter (mm) - 220 - 220

Generation Type Generate Formate Generate Formate

Cutting Method FH FH FH FH

Table 3.2. Basic design parameters of the face-milled spiral bevel gear set

Gear Set C

Parameter Pinion Gear

Number of Teeth 11 41

Hand of Spiral Left Right

Face Width (mm) 30.0 30.0

Mean Spiral Angle (deg) 33.7 33.7

Pinion Shaft Angle (deg) 90 -

Pinion Shaft Offset (mm) 0 -

Gear Pitch Diameter (mm) - 200

Generation Type Generate Formate

Cutting Method FM FM

Table 3.3. Simulation matrix for the parametric studies

Face- Hobbed 15 mm Offset 30 mm Offset Influences of

0.2 mm E error Shaft Offset Set A and Set B and 500 – 3000 rpm 0.2 mm P error Misalignments 80% Oil 0.2 mm G error

Face-Milled 0 mm Offset

1% Oil Influence of Set C Oil-to-Air Ratio 500 – 18000 rpm 3% Oil No Shaft Misalignments 5% Oil

3.3. Low-speed Simulations with High Oil-to-air Ratios

In this section, simulations representative of rear-axle applications are presented using the two sets of face-hobbed hypoid gears (Set A and Set B) with basic design parameters listed in

Table 3.1. Operating speed of the pinion simulates typical automotive applications and ranges from 500 rpm to 3000 rpm with increments of 100 rpm. To represent dip lubrication conditions, the oil-to-air ratio is selected as 80% ( ξ 0.80 . Influences of both shaft offset e and

misalignments on pocketing power loss are investigated. Table 3.3 summarizes the simulation

matrix used for the parametric studies in this section to investigate influences of shaft offset and

misalignments.

3.3.1. Influence of Shaft Offset

In order to quantify the influence of shaft offset, the gear sets used in this parametric study are tailored to have the same number of teeth and gear pitch diameters. Hence, the simulations more or less highlight the effects due to differences in pinion offset. Two values of shaft offset, 15 mm and 30 mm, are used and their effect on power loss due to

pocketing is quantified.

The influence of shaft offset on pocketing power loss, , is shown in Figure 3.1 within

the operating pinion speed range of Ω 500 3000 rpm to and at volumetric oil-to-air ratio of

ξ0.80. The variation in with speed is observed to be almost linear. The proportionality of

. . pocketing power loss to speed for 0.80 is ∝Ω for 15 mm and ∝Ω for

30 mm. The observed increase in with decreasing is relatively small in magnitude at

lower operating speeds. However, as Ω increases, the effects of e become more significant.

Figure 3.1. Variations in pocketing power loss, , with pinion speed, Ω, for face-hobbed hypoid gear pairs with shaft offsets, 15 and 30 , and no shaft misalignments operating under simulated dip lubrication conditions at oil-to-air ratio ξ 0.80

Looking at a decrease in shaft offset from 30 to 15 mm, is predicted to increase by about

33% when the volumetric oil-to-air ratio is 0.80. Yet, this percent increase corresponds to

a change of only ∆ 20 W in the pocketing power loss at Ω 3000 for the given gear

sets operating under the given conditions.

The increase observed in power loss due to pocketing is mainly because of the tightened

mesh with decreasing shaft offset. The active face width of gears decreases with smaller offset,

resulting in a decrease in the volume of fluid to be pumped out of the mesh. However,

simultaneously the mesh tightening shows a decrease in exit areas causing fluid velocity to

increase, eventually resulting in a modest increase of pocketing power loss with decreasing shaft

offset, as shown in Figure 3.1. It is also noted this trend is opposite of the one observed for the

mechanical power losses of hypoid gears as reported in earlier research by Xu and Kahraman [33]

and Kolivand et al [41] where both studies showed reduced mechanical power losses with

reduced shaft offset, e.

3.3.2. Influence of Shaft Misalignments

A set of parametric studies are performed to investigate the effect of misalignments on pocketing power loss for hypoid gear pairs with different values of shaft offset. Misalignments considered in this study are defined using the axes of the gear and pinion, as well as the shaft offset. Positive E error, δ, is taken as increasing shaft offset while positive G and P errors, δ

and δ, are taken along the positive direction of gear and pinion axes, as illustrated in Figure 3.2.

These errors were shown to influence contact patters, contact and root stresses as well as mechanical efficiency of a hypoid gear pair greatly [33, 41, 51]. The values of these misalignments are varied between 0.2 mm to investigate the sensitivity of to various

misalignments. These simulations are done for both gear sets A and B with 15 and 30 mm,

respectively, to observe the changes in the effects of misalignments with pinion offset.

Figure 3.2. Definition of shaft misalignments [52]

The influence of δ on is presented in Figure 3.3(a) for gear pair A in Table 3.1

15 mm within Ω 500 3000 rpm and at ξ 0.80. The effect of misalignment in the

direction of shaft offset is shown to be insignificant at lower speeds. As the operating speeds

increase, slight changes in predicted are observed with changing δ. Nevertheless, with the

operating speeds of consideration, the differences in power loss are still negligible samll. At oil-

to-air ratio 0.80, for a shaft misalignment of δ 0.2 , increased by 1.1%, whereas for

δ 0.2 mm, a decrease in of 0.3% is observed.

A similar, but slightly more significant effect is shown in Figure 3.3(b) demonstrating the influence of δ on for a gear pair B with 15 mm within the same speed range at the

same oil-to-air ratio. A misalignment of δ 0.2 mm increases by 2.9%, whereas for

0.2 mm, a 3.5% decrease in is observed. Although a higher percent change is

evident in power loss, the differences in magnitude are still insignificant due to low operating

speeds.

The influence of error on is given next in Figure 3.3(c) for the gear pair A with

15 mm, Ω 500 3000 rpm and ξ 0.80. It is observed that misalignments in either

direction results in a decrease of small percentage in power loss. A misalignment of δ 0.2

mm results in a 1.6% decrease in and δ 0.2 mm results in a 1.8% decrease in of

Similar to the effects of δ and δ errors, magnitudes of the changes in with are are also

found to be insignificant within the speed range of interest.

The same study is applied to the gear set B with 30 mm as well. As evident from the

results presented in Figure 3.4 under the same operating conditions, effects of misalignments δ,

δ and δ on are also found to be insignificant.

Figure 3.3. Variations in pocketing power loss, , with pinion speed, Ω, for a face-hobbed hypoid gear pair with shaft offset, 15 , and different values of shaft misalignments,

(a) 0.2 , (b) 0.2, and (c) 0.2 operating under simulated dip lubrication conditions at oil-to-air ratio ξ 0.80.

Figure 3.3 (continued)

Figure 3.4. Variations in pocketing power loss, , with pinion speed, Ω, for a face-hobbed hypoid gear pair with a shaft offset, 30 , and different values of shaft misalignments,

(a) 0.2 , (b) 0.2, and (c) 0.2 operating under simulated dip lubrication conditions at oil-to-air ratio ξ 0.80.

Figure 3.4 (continued)

3.4. High-speed Simulations with Low Oil-to-air Ratios

In this section, simulations of aerospace-like application are presented using a face-milled spiral bevel gear pair (Set C) with basic design parameters listed in Table 3.2. Operating speed range of the pinion up to Ω 18000 rpm is intended to represent the conditions for typical.

Three discrete values are selected for the oil-to-air ratio in the mixture as 1, 3 and 5% (ξ0.01,

0.03 and 0.05 to represent various jet lubrication flow conditions. The effect of increasing oil-to-

air ratio on pocketing power loss is studied at high rotational speeds. Table 3.3 summarizes the

simulation matrix used for the parametric studies in this section to investigate the influence of oil-

to-air ratio.

3.4.1. Influence of Oil-to-air Ratio

In order to quantify the effect of volumetric oil-to-air ratio of the fluid mixture, a set of parametric studies are performed at high operating speed conditions. A face-milled spiral bevel gear set (gear set C) with no shaft misalignments are used in these simulations.

Figure 3.5 shows the influence of volumetric oil-to-air ratio on pocketing power loss.

Despite low ratios of oil to air in the mixture for jet lubrication applications, power loss due to pocketing is predicted to be much higher than in dip lubrication applications under low speed conditions. This is a direct consequence of the higher operating speeds. Also, it is observed that sensitivity of pocketing power loss to changes in volumetric oil-to-air ratio increases at higher speeds. These conclusions are indicated by conservation of energy given in Equation (2.15c) in

Chapter 2, Section 2.3.4. Simplifying and expanding the control surface term gives the energy flux out of the control volume as

Figure 3.5. Variations in pocketing power loss, , with pinion speed, Ω, for a face-milled spiral bevel gear pair with no shaft misalignments, operating under simulated jet lubrication conditions at different values of oil-to-air ratios, ξ 0.01, ξ 0.03, and ξ 0.05

1 3.1 2

where is the fluid density, is the fluid exit velocity, is the exit area, is the exit pressure,

is the constant volume specific heat of the fluid and is the temperature. Note that

indicates integration over the control surfaces of a pocket. The energy flux in Equation (3.1) is

dominated by terms that are functions of since changes in in the third term are insignificant.

With increasing pinion operating speed Ω and corresponding increase in , the kinetic energy

term, , takes over the flow work term, , to dominate the energy flux single-handedly.

Also note that, the cubic fluid velocity term includes fluid density indicating that the effect of

density change, which is driven by changes in volumetric oil-to-air ratio, ξ, is more significant at

higher speeds. As a result, the relationship between and Ω is now an exponential function that

becomes more significant with increasing ξ as shown in Table 3.4 and Figure 3.5.

The increasing nonlinearity with oil-to-air ratio also results in an increase in power loss.

At Ω 18000 rpm, increasing oil in the fluid mixture from ξ0.01 to ξ 0.03 increased

by 18.5% , whereas increasing from 0.03 to 0.05 caused a 21.0% increase in . Since the

overall magnitudes of pocketing power loss are now sizable at high speeds, these percent changes

also correspond to larger changes of power loss in magnitude. Increasing the volumetric oil-to-

air ratio from ξ0.01 to 0.05 at Ω 18000 rpm is observed in a net pocketing loss increase of

∆ 0.2 kW.

Table 3.4. Proportionality of pocketing power loss, , to pinion speed, Ω, for Gear Set C at different values of oil-to-air ratios, ξ0.01, ξ 0.03, and ξ 0.05

Volumetric Oil-to-Air Ratio

Proportionality ξ0.01 ξ0.03 ξ0.05

. ∝Ω 500 – 13000 rpm 500 – 8000 rpm 500 – 6000 rpm

. ∝Ω 13000 – 18000 rpm 8000 – 13000 rpm 6000 – 11000 rpm

. ∝Ω - 13000 – 18000 rpm 11000 – 15000 rpm

. ∝Ω - - 15000 – 18000 rpm

3.3. Results and Discussions

In this chapter, results of several parametric studies to investigate the influences of

various parameters, including shaft offset, misalignments and oil-to-air ratio, on pocketing power

loss are presented. Simulations are done using the methodology introduced in Chapter 2, with two

face-hobbed hypoid gear pairs and a face-milled spiral bevel gear pair, replicating common

applications in automotive and aerospace industries. Hence, the capability of proposed model in

handling different types of cross-axis gears generated by different cutting techniques and

operating under various conditions is demonstrated.

Following conclusions can be drawn from the results parametric studies:

 Decreasing shaft offset causes an increase in power loss due to pocketing.

 Sensitivity of pocketing power loss to shaft misalignments is secondary regardless of

the shaft offset value.

 Increasing volumetric oil-to-air ratio causes an increase in pocketing power loss.

 Variation of pocketing power loss with speed becomes exponential with increasing

speed and oil-to-air ratio. A transition from a linear power loss-speed relation to

nonlinear is observed to occur at lower speeds with increasing oil-to-air ratio.

 Sensitivity of power loss due to pocketing to oil-to-air ratio increases with increasing

speed. Magnitude of power loss due to pocketing becomes more significant at higher

pinion speeds.

 In general, a higher operating speed magnifies the effect of other parameters

influencing pocketing power loss.

At a given speed, the magnitude of power loss due to pocketing is mainly affected by the steepness of the pressure gradient which is driven by the volumes and exit areas of individual pockets. Smaller exit areas cause the pressure gradient between pockets to be higher resulting in faster flow of fluid which increases the pocketing power loss. Spiral angle is one of the main parameters that influence the openness of exit areas. Thus, larger spiral angle in a gear pair results in larger exit areas, slower fluid flow and therefore less power loss due to pocketing [28].

As a result of geometry, mean spiral angles are typically large in hypoid and spiral bevel gears. When combined with the conical shape of the gears, the exit areas become even wider.

This results in less pressure gradient, slower flow of fluid between pockets and essentially lower pocketing power loss. Nevertheless, at high speeds, magnitudes of power loss due to pocketing can become significant as shown in the simulations presented in this chapter.

CHAPTER 4

CONCLUSION

4.1. Summary and Conclusions

A computational investigation of load-independent power losses due to fluid pocketing in

spiral bevel and hypoid gear pairs was conducted. The main goal of this research was to void the

gap in the gear efficiency literature in predicting pocketing losses of cross-axis gearing. For this

purpose, a model that combines a hypoid surface generation methodology with a fluid mechanics

formulation was developed to predict pocketing power losses.

As described in Chapter 2, the proposed model relied on a methodology that was

developed first by Kolivand [2] and later extended by Hotait [3] to define spiral bevel and hypoid

gears by simulation of the cutting processes. Based on the hypoidal coordinate frame introduced

by Hotait [3], a novel hypoidal discretization method was developed to define pockets of space

between the gear teeth in mesh along the face width and circumferential directions. Volumes of

each discrete pocket were defined along with the exit areas and associated centroids. Finally, a

physics-based fluid mechanics formulation developed by Talbot [28] that utilizes principles of

conservation of mass, conservation of momentum and conservation of energy was used to

compute load-independent power losses due to fluid pocketing.

In Chapter 3, various simulations and parametric studies were documented. With oil-to-

air ratio, shaft offset, speed and manufacturing method and shaft misalignments as variables, their

impact on pocketing losses were quantified. Overall, pocketing losses of hypoid gears operating

under the speed ranges of automotive axles were predicted to represent only a small portion of the

spin losses reported in earlier experimental studies [Xu etal, SAE and ASME papers, Hurley

thesis]. Power loss variations caused by misalignments, oil-to-air ratios as well as shaft off-set

were all found to be secondary as long as the speeds are relatively low. In contrast, at higher

rotational speeds representative of aerospace gearing, power losses were found to be significantly

higher with oil-to-air ratio becoming more significant.

4.2. Thesis Contributions

The primary objective and motivation behind this research was to fill the gap in the

literature of power losses in spiral bevel and hypoid gearing by developing a model to predict the

component of load-independent power losses due to fluid pocketing. With this objective

achieved, the following can be listed as contributions of this study :

 A hypoid surface generation methodology including the root geometry [2,3] was

combined first time with a fluid mechanics formulation [28] through a new discretization

scheme to predict pocketing power losses. The hypoidal discretization method

introduced in this study is novel and constitutes a marked improvement in computational

efficiency over traditional discretization methods used for parallel axis gears. This

discretization method uses lateral cone surfaces extending outwards to determine gear

and pinion section profiles.

 Pocketing power losses in spiral bevel and hypoid gear pairs were quantified at various

pinion operating speeds, shaft offset, misalignments and volumetric oil-to-air ratio in the

fluid mixture. This is the first study to quantify the influences of these parameters on

pocketing losses under typical automotive and aerospace operating conditions.

 With the proposed model in hand, measured spin losses of axles can be divided into their

pocking, face drag and bearing components. This provides the critical information

needed to devise strategies to minimize spin losses of such components.

4.3. Recommendations for Future Work

The following are listed as potential future research that may enhance or complement the

studies presented in this thesis:

 The proposed model can be extended to predict the power losses in an entire automotive

rear-axle or a rotorcraft transmission by implementing a complete thermal analysis of

these components.

 Studies on power losses due to fluid pocketing in spiral bevel gear pairs can be detailed

for rotorcraft applications by focusing on the magnifying effect of high-speed conditions.

Parametric studies can be performed to illustrate and investigate deeper the nonlinear

regions of pocketing power loss variation with speed.

 Current models computing windage/churning components of load-independent power

losses in spur gears [17,18] can be extended to spiral bevel and hypoid gears. The results

from the proposed model predicting load-independent power losses due to fluid

pocketing can be combined with these extensions and experiments can be performed to

verify the predicting capability of these models.

 The parametric studies presented in this thesis can be extended to cover the entire

spectrum of possible volumetric oil-to-air ratios. The simulations performed in this study

focused on replicating common applications in automotive and aerospace industries. It

was observed that increasing oil-to-air ratio in the fluid mixture increases the nonlinearity

of pocketing power loss variation with speed. This effect is obviously enlarged at higher

operating speeds.

 The proposed model, including the hypoid surface generation methodology [2] and fluid

mechanics formulation [28] used, can be extended and employed to crossed-helical,

worm and face gears.

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