AN EXPERIMENTAL INVESTIGATION OF HELICAL GEAR EFFICIENCY. A Thesis. Presented in Partial Fulfillment of the Requirements for

AN EXPERIMENTAL INVESTIGATION OF HELICAL GEAR EFFICIENCY A 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 Aarthy Vaidyanathan, B. Tech. ***** The Ohio State University 2009 Masters Examination Committee: Dr. Ahmet Kahraman Approved by Dr. Donald R Houser Advisor Graduate Program in Mechanical Engineering

ABSTRACT In this study, a test methodology for measuring load-dependent (mechanical) and loadindependent power losses of helical gear pairs is developed. A high-speed four-square type test machine is adapted for this purpose. Several sets of helical gears having varying module, pressure angle and helix angle are procured, and their power losses under jetlubricated conditions are measured at various speed and torque levels. The experimental results are compared to a helical gear mechanical power loss model from a companion study to assess the accuracy of the power loss predictions. The validated model is then used to perform parameter sensitivity studies to quantify the impact of various key gear design parameters on mechanical power losses and to demonstrate the trade off that must take place to arrive at a gear design that is balanced in all essential aspects including noise, durability (bending and contact) and power loss. ii

Dedicated to all those before me who had far fewer opportunities, and yet accomplished so much more. iii

ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Ahmet Kahraman, who has been instrumental in fostering interest and enthusiasm in all my research endeavors. His encouragement throughout the course of my studies was invaluable, and I look to him for guidance in all my future undertakings. I also wish to thank the sponsors, GM Powertrain, and Mr. Neil Anderson in particular for his support in carrying out this project. The members of the Gear Lab have been an integral part of the research I ve conducted. I would like to thank Mr. Jonny Harianto for generously lending his time and photography skills in order to capture great images of the experimental gear specimens and for his genuine concern for the students at gear lab. Mr. Sam Shon has always been a great source of information and has been at the forefront of efforts to ensure the safety of all the research associates. I thank him for his efforts on our behalf. I wish to also thank Dr. Sheng Li, who provided much needed counsel and shared his understanding of power transmission. I would also like to thank my friends and family for their confidence in my abilities, and for making sure that I had fun along the way. Special thanks to all the men and women in the military and in various other areas of service to society, who keep the nations of the world safe from myriad problems, allowing us to focus on research and development. Their selfless service is the source of my strength and inspiration. iv

VITA December 31, 1985.. Born Trichy, India. August 2003 May 2007.. BTech Mechanical Engineering National Institute of Technology Surathkal, India. September 2007 June 2009.. Graduate Research Associate Department of Mechanical Engineering The Ohio State University, Columbus, Ohio PUBLICATIONS 1. S. Li, A. Vaidyanathan, J. Harianto and A. Kahraman, 2009, Influence of Design Parameters and Micro-geometry on Mechanical Power Losses of Helical Gear pairs, JSME International Conference on Motion and Power Transmissions, Sendai, Japan. Major Field: Mechanical Engineering FIELDS OF STUDY v

TABLE OF CONTENTS Page ABSTRACT... ii DEDICATION... vi ACKNOWLEDGMENTS... iv VITA... v LIST OF TABLES... x LIST OF FIGURES... xi 1 INTRODUCTION... 1 1.1 Introduction... 1 1.2 Literature Review... 2 1.3 Thesis Objectives... 5 1.4 Thesis Outline... 6 2 HELICAL GEAR EFFICIENCY TEST METHODOLOGY... 8 vi

2.1 Introduction... 8 2.2 Helical Gear Efficiency Test Machine... 8 2.3 Helical Gear Power Loss Test Procedure... 12 2.4 Helical Efficiency Test Gear Specimens and Test Matrix... 13 2.5 Specimen Consistency... 22 2.6 Repeatability of the Torque Measurements... 30 2.7 Summary... 35 3 HELICAL GEAR POWER LOSS MEASUREMENTS... 36 3.1 Introduction... 36 3.2 Determining the Components of Power Loss and Efficiency from Measurements... 36 3.3 Gearbox Mechanical Power Loss Results... 40 3.3.1 Influence of Torque and Speed... 41 3.3.2 Influence of Helix and Pressure Angles... 46 vii

3.4 Gearbox Spin Power Loss Results... 56 3.5 Conclusion... 58 4 COMPARISON OF EXPERIMENTAL RESULTS TO PREDICTIONS OF A GEAR EFFICIENCY MODEL... 59 4.1 Introduction... 59 4.2 Gear Mesh Mechanical Power Loss Model... 60 4.2.1 A Mixed EHL Model to Predict the Friction Coefficient... 61 4.2.2 Derivation of New Friction Coefficient Formula for Gear Contacts.. 65 4.3 Comparison of Measured and Predicted Power Loss Values... 71 4.4 Conclusions... 97 5 PARAMETER SENSITIVITY STUDY... 98 5.1 Introduction... 98 5.2 Analysis Details... 100 5.3 Variation of the Gear Mesh Mechanical Power Loss with Gear Design Parameters... 103 viii

5.4 Selection of an Optimum Design Satisfying Multiple Performance Criteria 105 5.5 Conclusions... 116 6 SUMMARY AND CONCLUSIONS... 117 6.1 Summary... 117 6.2 Conclusions... 118 LIST OF REFERENCES... 120 ix

LIST OF TABLES Table Page 2.1 Test matrix of the helical gear efficiency experiments.... 15 2.2 Basic design parameters of the test gears.... 20 2.3 Test sequence used in the helical gear efficiency experiments.... 23 2.4 Measured average surface roughness parameters of gear test specimens.... 33 3.1 Values of parameters for the bearings used in the experiments.... 39 4.1 Parametric design for the development of a friction coefficient formula... 67 4.2 Basic parameters of the automatic transmission fluid used in this study... 68 5.1 Values of the gear parameters varied in the parameter sensitivity study.... 99 5.2 Gear parameters of the four different designs identified in Figure 5.3.... 109 x

LIST OF FIGURES Page 2.1 Helical gear efficiency test machine.... 9 2.2 Schematic representation of the helical gear efficiency test machine.... 10 2.3 An example variation of (a) T L and (b) lubricant temperatures at various locations as a function of time from a test run at Ω = 6000 rpm, and T c = 413 Nm.... 14 2.4 Pictures of the gear test specimens used in the study; (a) gear set A, (b) gear set B, (c) gear set C, (d) gear set D, (e) gear set E, (f) gear set F, (g) gear set G, (h) gear set H, and (i) gear set I.... 17 2.5 A test gear set being inspected on the gear CMM.... 24 2.6 Example gear profile traces of the test gears; (a) gear A, (b) gear B, (c) gear C, (d) gear D, (e) gear E, (f) gear F, (g) gear G, (h) gear H, and (i) gear I. (Continued)... 25 2.7 Measurement of surface roughness using the Taylor-Hobson Form Talysurf- 120 surface profiler with the aid of a special mounting fixture.... 31 2.8 Sample surface roughness profiles of the helical gear efficiency test gears (a) 23T, Ψ = 30, Φ = 22.7 (Set C) and (b) 40T, Ψ = 20, Φ = 17.8 (Set H).... 32 xi

2.9 Demonstration of the repeatability of the test setup through comparison of measured T L values of tests on gear set A (23T, Ψ = 0, Φ = 25.9 ) at various conditions (refer to Table 2.3 for test conditions).... 34 3.1 Measured (a) P mech and (b) ηmech values of the 23-tooth gear pair B having Ψ = 30 and Φ = 25.9 at various torque values.... 42 3.2 Measured (a) P mech and (b) η mech values of the 40-tooth gear pair I having Ψ = 21 and Φ = 16.5 at various torque values.... 43 3.3 Measured (a) P mech and (b) η mech values of the 23-tooth gear pair B having Ψ = 30 and Φ = 25.9 at various speed values.... 44 3.4 Measured (a) P mech and (b) ηmech values of the 40-tooth gear pair I having Ψ = 21 and Φ = 16.5 at various speed values.... 45 3.5 Measured P mech value of the 23-tooth gear sets A to C at (a) T c = 140 Nm, (b) T c = 239 Nm, (c) T c = 413 Nm, and (d) 546 Nm.... 47 3.6 Measured η mech value of the 23-tooth gears A to C at (a) T c = 140 Nm, (b) T c = 239 Nm, (c) T c = 413 Nm, and (d) T c = 546 Nm.... 49 3.7 Measured P mech value of the 40-tooth gears D to I at (a) T c = 140 Nm, (b) T c = 239 Nm, (c) T c = 413 Nm, and (d) T c = 546 Nm.... 52 3.8 Measured η mech value of the 40-tooth gears D to I at (a) c140 Nm, (b) T c = 239 Nm, (c) T c = 413 Nm, and (d) T c = 546 Nm.... 54 xii

3.9 Measured P spin values of (a) 23-tooth gear sets A to C and (b) 40-tooth gear sets D to I.... 57 4.1 The relationship between the friction coefficient µ and lambda ratio λ.... 70 4.2 Comparison of predicted and measured P mesh values of 23-tooth gear pair A (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm....72 4.3 Comparison of predicted and measured P mesh values of 23-tooth gear set B (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000rpm....74 4.4 Comparison of predicted and measured P mesh values of 23-tooth gear pair C (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm....76 4.5 Comparison of predicted and measured P mesh values of 40-tooth gear pair D (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm....78 4.6 Comparison of predicted and measured P mesh values of 40-tooth gear set E (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm....80 4.7 Comparison of predicted and measured P mesh values of 40-tooth gear set F (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm....82 4.8 Comparison of predicted and measured P mesh values of 40-tooth gear set G (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm....84 4.9 Comparison of predicted and measured mesh P values of 40-tooth gear set H (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm....86 xiii

4.10 Comparison of predicted and measured P mesh values of 40-tooth gear set I (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm....88 4.11 Variation of P mesh of test gear pairs with normal pressure angle Φ at Ω= 2000 rpm and (a) T c = 140 Nm and (b) T c = 546 Nm. (a1, b1) predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs....91 4.12 Variation of P mesh of test gear pairs with helix angle Ψ at Ω= 2000 rpm and (a) T c = 140 Nm and (b) T c = 546 Nm. (a1, b1) predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs....92 4.13 Variation of P mesh of test gear pairs with module m at Ω= 2000 rpm and (a) T c = 140 Nm and (b) T c = 546 Nm. (a1, b1) predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs....93 4.14 Variation of P mesh of test gear pairs with profile contact ratio ε p at Ω= 2000 rpm and (a) T c = 140 Nm and (b) T c = 546 Nm. (a1, b1) predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs....94 4.15 Variation of P mesh of test gear pairs with face contact ratio ε f at Ω= 2000 rpm and (a) T c = 140 Nm and (b) T c = 546 Nm. (a1, b1) predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs....95 4.16 Variation of P mesh of test gear pairs with total contact ratio ε t at Ω= 2000 rpm and (a) T c = 140 Nm and (b) T c = 546 Nm. (a1, b1) predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs....96 xiv

5.1 Variation of P mesh with (a) N, (b) m, (c) d o, (d) ε f, (e) ε p, and (f) ε t...101 5.2 Variation of P mesh with (a) Φ, (b) Ψ, and (c) Λ wd...106 5.3 Variation of P mesh with (a) peak-to-peak TE, (b) maximum σ c, and (c) maximum σ b...110 5.4 Variation of P mesh [kw] with M P and M L at Tin values of (a) 50 Nm, (b) 250 Nm and (c) 500 Nm.... 112 5.5 Variation of TE [ µ m] with M P and M L at T in values of (a) 50 Nm, (b) 250 Nm and (c) 500 Nm.... 113 5.6 Variation of σ c [ MPa ] with M P and M L at T in values of (a) 50 Nm, (b) 250 Nm and (c) 500 Nm.... 114 5.7 Variation of σ b [ MPa ] with M P and M L at T in values of (a) 50 Nm, (b) 250 Nm and (c) 500 Nm.... 115 xv

CHAPTER 1 INTRODUCTION 1.1 Introduction Power losses of gear systems and transmission are becoming an increasingly important research topic as environmental regulations become more stringent and energy concerns become more critical. Increased gear efficiency helps improve fuel economy of a vehicle while reducing carbon emissions. It also reduces the demand on the lubrication systems and affects most of the failure modes such as scuffing and pitting positively. Power losses of a gear pair are influenced by a large number of factors including contact surface parameter, lubrication method and lubricant parameters, and operating speed, load and temperature conditions as well as various gear design parameters. This study focuses on load-dependent (mechanical) power losses and efficiency of helical gear pairs. It aims at establishing a helical gear efficiency database, and using it to validate a gear power loss model developed in a companion study by Li et al. [1]. The model is then used to study the study the influence of basic design parameters and tooth surface modifications on the mechanical power losses of a helical gear pair, and in the selection 1

of a design that satisfies multiple performance characteristics like noise and contact stress. 1.2 Literature Review The power losses in a gearbox are comprised of two types of losses the loaddependent and the load-independent (spin) losses. The experimental portion of this study deals with both types of losses, while the model comparison and parametric study focus on the load-dependent component of power loss. The spin consist of windage and churning losses and are associated with fluid-gear interactions [2, 3], while sliding and rolling friction loss contribute to the load-dependent losses. Literature on gear power losses and efficiency consisted of studies that took different approaches to defining a friction coefficient. Some, as in reference [4, 5] employed a simple model with a uniform sliding friction coefficient. This approach, although useful as a preliminary model in understanding the effects of gear geometry on power loss and efficiency, did not account for various parameters that affect a typical gear contact. A number of experimental investigations like those by Naruse et al. [6], Petry-Johnson [7], and Moorhead [8] have indicated that parameters like sliding and rolling velocity, lubricant properties like viscosity, the load, and surface roughness have a significant effect on the friction coefficient. Other studies [9-11] included the effects of these parameters to a limited extent utilizing empirical friction relations derived from curve fitting of twin-disk type traction measurements. These models, while providing a better prediction than the constant friction model, did not accurately predict the power 2

loss in a gear mesh, as they were derived based on data collected through experiments on twin-disk type machines not specifically designed to represent gear tooth contact. Xu et al. [12] provided a detailed review of these earlier gear efficiency studies, indicating that the friction coefficient varies considerably along the tooth surface, and that the accuracy of the empirical friction models depend on the experimental conditions that the adapted friction formula was based on. A third approach was to include the elastohydrodynamic lubrication (EHL) behavior in the model directly, as in the spur gear models developed by Dowson and Higginson [13] and Martin [14]. Xu et al. [12] also provided an EHL based methodology for prediction of gear mechanical efficiency using the model by Cioc et al. [15] to predict the friction coefficient for each combination of the contact parameters considered, and applied a linear regression analysis to derive a single friction coefficient formula. The formulation included important parameters like rolling and sliding velocities, radii of curvature, lubricant viscosity (temperature) and surface roughness amplitudes. In tandem with a load distribution model [16], the formula was used to predict the mechanical power losses of the spur gears used in the experiments of Pertry- Johnson et al. [7], and the predictions were found to match reasonably well with the experiments. As the EHL model of Cioc et al. [15] allowed only a few asperity contacts, the model of Xu et al. [12] was limited to gear contacts with limited asperity interactions. Most automotive gears have reasonably large surface roughness values originating from shaving or grinding processes and operate under low speed, heavy load and high 3

temperature conditions with lubricants that are suboptimal, such that mixed EHL conditions with severe metal-to-metal contacts are common. The new mixed EHL model developed by Li and Kahraman [17] was well suited for these conditions with asperity contacts. Experimental work in gear efficiency is relatively smaller in volume compared to the theoretical studies. Naruse et al. [6] carried out a series of experimental studies investigating the effect of load and rotational speeds on efficiency, and the difference in efficiency between ground and shaved gears. Xiao et al. [18] and Britton et al. [19] studied traction measurements and concluded that lower values of surface roughness decrease gear mesh power losses. The material in the literature dealing with studies on spur gear efficiency was large compared to the relative lack of studies on helical gear efficiency. The study by Handschuh and Kilmain [20] dealt with preliminary experimental and analytical efficiency results of high speed helical gear trains. However, it used a helical gear train that was representative of a prop-rotor gearbox of a tilt-rotor aircraft and its associated specific operating conditions. The results also focused more on windage losses. Heingartner and Mba [21] studied the power losses in the helical gear mesh, but used a coefficient of friction that was independent of gear surface temperature, a parameter that is of significant importance. The investigation was aimed at the effects of increasing load with constant speed, and concluded that the sliding friction losses were heavily load dependent where as rolling friction losses decreased slightly with increased load. 4

A series of spur gear efficiency experiments were conducted under both jet lubricated and dip lubricated conditions in The Ohio State University. A test methodology was put in place [22] by means of which efficiency measurements could be made at high speeds and loads of up to 10,000 rpm and 700 Nm respectively, under jet lubricated conditions. Petry-Johnson [7] and Moorhead [8] continued this study. They studied the effects of gear module, face width, surface roughness, manufacturing process, and lubrication oil on both load-dependent and load-independent gear power losses. Moorhead [8] expanded the experimental database by performing tests under conditions that were more representative of automotive transmission applications. Non-unity ratio spur gears of medium quality and common lubricants were used. Moorhead also carried out experimental studies to catalogue churning losses under dip-lubrication conditions. All of the tests were carried out on spur gears, until a helical gear efficiency test machine was developed by Moorhead [8] to enhance the testing capability to include helical gears. 1.3 Thesis Objectives The main objectives of this study are defined below: Develop a test methodology to measure the power losses of spur and helical gears at high-power conditions on the new helical gear efficiency test machine developed by Moorhead [8]. Establish a helical gear efficiency database through tightly controlled helical gear efficiency experiments. 5

Describe the trends in power losses in a helical gear mesh based on gear design parameters under various operating conditions. Compare the experimental results with the predictions of a gear efficiency model developed by Li et al. [1] and evaluate the accuracy of the model. Employ the model [1] to identify and quantify the influence of key gear design parameters impacting efficiency. Demonstrate the simultaneous influence of the gear design parameters on efficiency, transmission error and gear stresses and obtain well balanced designs that are acceptable in all aspects (efficiency, durability and noise), with some compromise in each. Refine the good design solutions identified in order to quantify the influence of micro-geometry on the gear stresses, noise metrics and mechanical efficiency. 1.4 Thesis Outline The experimental set up and the helical gear efficiency test methodology is described in Chapter 2, along with a description of the instrumentation used in the tests. Test specimens are described in detail along with the test matrix, inspection procedures undertaken to ensure surface consistency of the gears throughout the tests and the repeatability studies performed. Chapter 3 presents the results of helical gear power loss tests carried out under jet lubricated conditions in the order defined in the test matrix in 6

Chapter 2. The results are presented in terms of gearbox mechanical and spin power losses to quantify the influence of speed, load, module, pressure angle, and helix angle. In Chapter 4, the experimental results of the helical gear efficiency tests are compared with the predictions of the gear efficiency model developed by Li et al. [1]. The values of power loss in the gear mesh are then presented as a function of key gear parameters. Based on the comparisons, the accuracy of the helical gear pair efficiency model is discussed in the end. In Chapter 5, a sample parameter study is undertaken, and the results are used to quantify the impact of key design parameters on gear pair mechanical power loss. Based on this study, a candidate design that balances all critical gear performance requirements (noise, durability and efficiency) is used to perform the microgeometry study to show the influence of tooth surface modifications. Finally, the major conclusions of this study are listed in Chapter 6. 7

CHAPTER 2 HELICAL GEAR EFFICIENCY TEST METHODOLOGY 2.1 Introduction In this chapter, the experimental set up and test methodology used to measure helical gear pair power losses is described in detail. The test machine and the test specimens are introduced, and test, measurement and inspection procedures are described. At the end, repeatability and resolution of the measurements are discussed. 2.2 Helical Gear Efficiency Test Machine The high-speed helical gear efficiency test machine used in this study was designed by Moorhead [8] during the spur gear efficiency phase of this project. Details of the test machine and the data acquisition and processing system are available in references [7, 22, 23]. Figure 2.1 shows a picture of the helical gear efficiency test machine used in this study and Figure 2.2 shows a schematic representation of the same. 8

Figure 2.1 Helical gear efficiency test machine. 9

Flexible Coupling Precision Torquemeter Belt Drive Input Shaft High Speed Spindle Torque Test Side gearbox Reaction Side gearbox AC Motor Split Coupling Figure 2.2 Schematic representation of the helical gear efficiency test machine. 10

The machine consists of a pair of opposing, identical gear boxes, each containing a pair of unity-ratio helical (or spur) gears supported by rigid shafts mounted on two pairs of bearings. They are labeled as Test Gearbox and Reaction Gearbox in Figure 2.2 for the purpose of distinguishing them while they are identical in every aspect. The assembly procedure for gears is the same as the one used in earlier spur gear efficiency tests [7, 22, 23], and is described in greater detail in reference [8]. Each gear is held in place between two bearings, a deep-groove ball bearing on the outside (SKF Model 6206) and a four-point angular contact ball bearing (SKF Model QJ 206 MA) on the inside. The deep-groove ball bearing is not axially loaded (allowed to float axially), allowing all of the axial load to be carried by the four-point angular contact ball bearing. In this arrangement, the bearings are not required to be preloaded. This avoids repeatability problems associated with setting the correct amount of preload for each test as well as changes in preload due to temperature effects that would be detrimental to the fidelity of the measurements. In Figure 2.2, two gear pairs are connected to each other through flexible shafts. A split coupling is mounted on one of the shafts connecting the gearboxes and is used to apply a constant torque T c to the closed loop [7]. The gearboxes are powered by a variable speed AC motor connected to a high-speed spindle. The connection is through a 3:1 ratio belt drive. The high speed spindle is connected to the splined input shaft of the reaction gearbox through a flexible coupling. The gearboxes and shafts connecting them are mounted on a sliding table, allowing easy engagement and disengagement of the splined input shaft from the coupling. The torque loss in the system T L is measured 11

using a high-precision, non-contact type torque-meter (Lebow model TMS 9000) that is mounted between the high-speed spindle and the input shaft to the reaction gearbox. The temperature control is achieved through the lubrication system that supplies the lubricant to the gear meshes and bearings through nozzles. The oil temperature is measured at various points (in the lubricant heating unit, and the supply and return points) on the test machine, which is then used to control the lubricant temperature to a preset value. The lubrication systems and parameters of the test and reaction gearboxes are kept identical to ensure the same thermal conditions for both gearboxes. Likewise, oil flow rates are tightly controlled and monitored as well. The maximum operating speed that the test machine can reach is 10,000 rpm, which corresponds to a pitch line velocity of 48 m/s for unity ratio gears with a fixed center distance of 91.5 mm. The machine controls and monitoring of control and measurement parameters are the same as those used in previous studies on a spur gear test machine, as described by Petry-Johnson [7]. 2.3 Helical Gear Power Loss Test Procedure All the helical gear efficiency tests in this study were conducted by using a typical automatic transmission fluid (ATF) as the lubricant, provided at a fixed temperature of 90 C. A Labview interface is used to monitor the temperature as well as set the test duration and rotational speed. A calibrated jet lubrication system was used to deliver controlled amounts of oil to gear meshes and bearings, similar to the one used in the spur gear studies conducted previously [7, 22]. 12

The new test gears were first run-in at a low torque ( T c = 140Nm) and speed ( Ω= 2000 rpm) values for a duration of one hour. The actual duration of each test was kept at 11 minutes [7, 8, 22, 23]. Figure 2.3(a) shows an example T L as a function of time, measured at a test condition of Ω = 6000 rpm, and Tc = 413 Nm. Here, the transient behavior is typically eliminated after the first 5 to 6 minutes as it was the case in previous spur gear power loss measurements on the same machine [7, 8, 22, 23]. The torque measured by the torque-meter was the total torque provided to the closed loop by the motor in order to maintain operation, and represented the torque loss of the entire closed loop. After completion of the test, any electronic drift of the torque-meter was recorded, to be subtracted from the torque loss value obtained. Prior to conducting the experiments, the lubrication system was set at the testing temperature. The system was allowed to stabilize for an hour before testing. Prior to each test, it was ensured that the lubricant levels in the lube sump were above the minimum required levels for a complete sequence of tests. Figure 2.3(b) shows an example variation of temperature as a function of time for the same test used in Figure 2.3(a). 2.4 Helical Efficiency Test Gear Specimens and Test Matrix Nine different gear sets were used in this study to implement the test matrix shown in Table 2.1. This test matrix was formed jointly with the sponsor with the goal of (i) investigating the influence of basic helical gear design parameters, namely normal pressure angle Φ, helix angle Ψ and normal module m, on gear mesh power losses, and 13

T L [ Nm] 10 9 8 7 6 5 4 3 2 1 0 120 Torque loss (a) 0 2 4 6 8 10 12 (b) 100 Temp [ C] 80 60 40 20 0 Temperature, Sump Test Supply Reaction Supply Test Return Reaction Return 0 2 4 6 8 10 12 Time [minutes] Figure 2.3 An example variation of (a) T L and (b) lubricant temperatures at various locations as a function of time from a test run at Ω = 6000 rpm, and T c = 413 Nm. 14

Test No. Gear Set Number of Ψ Φ Name Teeth [deg] [deg] 1 A 23 0 25.9 2 B 23 30 25.9 3 C 23 30 22.7 4 A Repeatability 5 D 40 0 26.5 6 E 40 30 26.5 7 F 40 30 23.4 8 F Repeatability 9 G 40 17.5 20.2 10 H 40 20 17.8 11 I 40 21 16.5 12 G Repeatability Table 2.1 Test matrix of the helical gear efficiency experiments. 15

(ii) validate a helical gear pair mechanical power loss model developed in a companion study. Figure 2.4 shows pictures of the various test gears and Table 2.2 lists the basic design parameters of each test gear pair. All the gears had the same face width of 26.67 mm and the center distance was fixed at 91.5 mm. All of the gears were hard ground to precise dimensions. According to the test matrix of Table 2.1, tests were performed in three groups. The first group of tests formed by unity-ratio, 23-tooth gear sets A, B and C. Here, gear set A was formed by 23-tooth spur gears ( Ψ= 0 ) with a pressure angle of Φ= 25.9. Gear set B with normal pressure angle of Φ= 25.9 and helix angle of Ψ= 30 was a helical equivalent of gear set A with the same normal pressure angle. Meanwhile, gear set C with Φ= 22.7 and Ψ= 30 had the same transverse geometry (transverse pressure angle and tooth thickness) as gear set A. The second group in the test matrix was formed by unity-ratio, 40-tooth test gears with the first one in the group (gear set D) being a spur gear pair with Φ= 26.5. Similar to first group, gear set E with normal pressure angle of Φ= 26.5 and Ψ= 30 was a helical equivalent of gear set D with the same normal pressure angle, while gear set F with Φ= 23.4 and Ψ= 30 had the same transverse geometry as gear set D. The first two groups of gear specimens were based on previous spur gear experiments [7, 8, 22, 23] as spur gears A and D were borrowed from these previous studies. The helical gears in these groups were designed based on these two spur gears. 16

(a) (b) (c) Figure 2.4 Pictures of the gear test specimens used in the study; (a) gear set A, (b) gear set B, (c) gear set C, (d) gear set D, (e) gear set E, (f) gear set F, (g) gear set G, (h) gear set H, and (i) gear set I. (Continued) 17

Figure 2.4 Continued. (d) (e) (f) (Continued) 18

Figure 2.4 Continued. (g) (h) (i) 19

Parameter A B C D E Number of Teeth 23 23 23 40 40 Normal Module 3.95 3.45 3.42 2.32 1.98 Normal Pressure Angle 25 25.9 22.7 26.5 26.5 (Operating) [deg.] Helix Angle [deg.] 0 30 30 0 30 Base Diameter 82.34 79.85 82.34 81.89 79.30 Major Diameter 100.25 100.25 100.25 95.88 95.88 Minor Diameter 81.36 81.36 81.23 85.72 85.72 Tooth thickness 6.43 6.16 6.422 2.92 3.51 (Transverse) Backlash 0.195 0.125 0.129 0.189 0.121 (Transverse) Roll angle at tip 43.49 39.79 38.95 chamfer [deg.] Profile Crown 0.012 0.012 0.012 Tip Relief 0.07 0.07 Tip Relief start [deg.] 36.0 32.5 Lead Crown 0.008 0.006 0.006 0.008 0.006 Table 2.2 Basic design parameters of the test gears. All dimensions are in mm unless specified. 20 (Continued)

Table 2.2 Continued. Parameter F G H I Number of Teeth 40 40 40 40 Normal Module (mm) 2.01 2.18 2.15 2.14 NormalPressure Angle 23.40 20.22 17.75 16.55 (Operating) [deg.] Helix Angle [deg.] 30.00 17.5 20.00 21.00 Base Diameter 81.89 85.36 86.61 87.19 Major Diameter 95.88 95.95 95.95 95.88 Minor Diameter 85.72 85.70 85.34 85.22 Tooth thickness 2.91 3.51 3.51 3.51 (Transverse) Backlash 0.125 0.121 0.123 0.124 (Transverse) Roll angle at tip 34.90 29.42 10.612 26.21 chamfer [deg.] Profile Crown 0.008 0.010 0.010 0.012 Tip Relief Tip Relief start [deg.] Lead Crown 0.006 0.006 0.006 0.006 21

The 40-tooth helical gears in the third group, meanwhile, were designed by the sponsor to represent typical automotive transmission final drive helical gears. Gear set G had Φ= 20.2 and Ψ= 17.5, gear set H had Φ= 17.8 and Ψ= 20, while gear set I had Φ= 16.5 and Ψ= 21. Each test listed in Table 2.1 was repeated at discrete speed values of Ω= 2,000, 4,000 and 6,000 rpm and torque levels of T c = 0, 140, 236, 413 and 546 Nm. Here, the tests at T c = 0 served a special purpose as they represented the load-independent spin losses. The measured losses from these unloaded tests were subtracted from a loaded test at the same speed value to isolate the load dependent (mechanical) losses. Table 2.3 specifies the sequence at which each test was performed. Three additional tests were performed as repeat of tests with gear set A, D and G as listed in Table 2.1 as tests 4, 8 and 12. These test results were used to establish the repeatability of the tests in each of the three test groups. 2.5 Specimen Consistency As the mechanical gear mesh power losses were of special interest, the tooth profile and lead accuracy of the test gears and the actual surface roughness amplitudes were verified before each test. As in previous studies [7, 8, 22, 23], a Gleason M&M 255 gear CMM was used to measure gear lead and profiles as well as pitch, indexing and spacing errors. Figure 2.5 shows a test gear being inspected on the M&M machine and Figure 2.6 shows the profile traces of one gear from each of the nine gear 22

T c Ω [rpm] [Nm] 2000 4000 6000 0 13 14 15 140 1 2 3 239 10 11 12 413 4 5 6 546 7 8 9 Table 2.3 Test sequence used in the helical gear efficiency experiments. 23

Figure 2.5 A test gear set being inspected on the gear CMM. 24

(a) (b) Figure 2.6 Example gear profile traces of the test gears; (a) gear A, (b) gear B, (c) gear C, (d) gear D, (e) gear E, (f) gear F, (g) gear G, (h) gear H, and (i) gear I. (Continued) 25

Figure 2.6 Continued. (c) (d) (Continued) 26

Figure 2.6 Continued (e) (f) (Continued) 27

Figure 2.6 Continued. (g) (h) (Continued) 28

Figure 2.6 Continued. (i) 29

pairs. Here it is noted that the test gears are of high quality. The gear CMM inspections were repeated after the tests were completed as well, to demonstrate that the changes to tooth profiles due to surface wear are negligible. The surface roughness profiles of each test gear in the involute direction were also measured by using a Taylor-Hobson Form Talysurf-120 surface profiler with the aid of a special gear mounting fixture. Figure 2.7 shows a test gear as being inspected for its surface roughness on this roughness profiler, while Figure 2.8 shows examples of measured run-in roughness profiles for both 23-tooth and 40-tooth gears. In addition, run-in average centerline roughness R a and the root-mean-square roughness Rq values for all nine test gear pairs are listed in Table 2.4. As in previous spur gear studies, these values were obtained for one tooth, on both the left and right flanks for each gear used in this study. The average values presented in Table 2.4 are those of all gears of the same type used in a test. The gears indicated a slight smoothing effect after run-in, but remain consistent throughout the rest of the tests. 2.6 Repeatability of the Torque Measurements As in any measurement, repeatability of the tests is of primary concern here, especially considering that the measured T L values are rather low and the differences between the tests are typically narrow. In order to ascertain the repeatability of the experimental data and ensure the consistent performance of the various fixtures, a complete test with all 15 operating conditions was repeated in each group of tests listed in 30

Figure 2.7 Measurement of surface roughness using the Taylor-Hobson Form Talysurf-120 surface profiler with the aid of a special mounting fixture. 31

2 (a) 0 Surface Roughness [ µm] -2 20.1 0.4 0.7 1.0 (b) 0-2 0.1 0.4 0.7 1.0 Distance [ mm] Figure 2.8 Sample surface roughness profiles of the helical gear efficiency test gears (a) 23T, Ψ = 30, Φ = 22.7 (Set C) and (b) 40T, Ψ = 20, Φ = 17.8 (Set H). 32

Gear Set R a R q [ µ m ] [ µ m ] A 0.28 0.37 B 0.22 0.28 C 0.24 0.32 D 0.21 0.28 E 0.23 0.29 F 0.31 0.39 G 0.21 0.27 H 0.26 0.34 I 0.26 0.34 Table 2.4 Measured average surface roughness parameters of gear test specimens. 33

T L [ Nm] 10 9 8 7 6 5 4 3 2 1 0 Test Repeat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Test Condition Figure 2.9 Demonstration of the repeatability of the test setup through comparison of measured T L values of tests on gear set A (23T, Ψ = 0, Φ = 25.9 ) at various conditions (refer to Table 2.3 for test conditions). 34

Table 2.1. As these repeat tests were performed several days after the original test with several other tests in between, they represent true repeatability of the entire setup. The procedure for ensuring consistency of test specimens was described in the previous section. The accuracy of the auxiliary components of the test machine were dealt with in detail by Petry-Johnson [7]. The accuracy of the torque loss TL read by the torque-meter was specified by the vendor as 0.03% of full scale range (50 Nm). Figure 2.9 compares two tests performed using gear set A having Φ= 25.9 and Ψ= 0. Here, the measured T L values are repeated consistently well within 0.25 Nm (corresponding to gearbox power loss repeatability within 0.05 kw). This can be deemed acceptable for the tests considered in this study. 2.7 Summary In this chapter, the test machine and instrumentation used for spur and helical gear power losses has been introduced. Test methodology, test matrix and the gear test specimens have been described. The results have been presented to demonstrate the repeatability of the measurements. 35

CHAPTER 3 HELICAL GEAR POWER LOSS MEASUREMENTS 3.1 Introduction The influence of the basic design parameters including pressure and helix angles as well as the rotational speed and input torque on load dependent (mechanical) and loadindependent (spin) power losses of helical gear pairs are studied experimentally in this chapter. Applying the test methodology proposed in the previous chapter, the experiments defined in the test matrix of Table 2.1 are executed under both loaded and unloaded conditions defined by Table 2.3 in order to separate the load independent losses from the measured total gear pair power loss. The bearing power losses are estimated using simplified bearing loss formulae [24] and subtracted from the mechanical losses to obtain the gear mesh power loss. 3.2 Determining Components of Power Loss and Efficiency from Measurements Under loaded condition, the rolling and sliding friction in both the gear and bearing components contribute to the power losses that are categorized as the load dependent or mechanical losses. The windage and churning losses of the gear pair, which 36

are mostly dependent on the geometry, speed, lubricant viscosity and lubrication method (dip or jet lubricated) as well as the load independent viscous friction loss in the rolling element bearings, also contribute to the power losses. These losses are referred to as the spin losses. With these, the total power loss P T of a gearbox can be written as PT = Pmech + Pspin, (3.1) where P mech is the total mechanical power loss of the gearbox including both bearings and the gear mesh and P spin is the total spin power loss. A two-step measurement procedure is used here in order to determine the mechanical power losses. At a certain rotational speed Ω, the experiment is first performed under the loaded condition with a specified T c. With the assumption that both test and reaction gearboxes are identical, the measured torque loss T L can be divided by two to find the power loss of an individual gearbox, from which he total gearbox power loss is found as PT = 1 ω T 2 L, (3.2) where ω = 2πΩ 60. Then the same test is repeated with zero loop torque ( T = 0) such that the measured T L overcomes the load-independent losses, i.e. Pmech 0 and 1 T spin = ω L. These two companion tests allow the measurement of P T = 0 mech P P T 2 c according to Eq. (3.1). c 37

Each gearbox in Figure 2.2 contains one gear pair, two identical four-point angular contact bearings (for both axial and radial load support) and two identical deepgroove ball bearings (for radial load support only). Hence, the mechanical power loss of each gearbox can be further decomposed into three parts as Pmech Pmesh 2Pb 1, L 2Pb 2, L = + + (3.3) where P mesh is the gear mesh power loss, and P b1, L and P b2, L are the load dependent power losses of a four-point angular contact bearing and a deep-groove ball bearing, respectively. Load dependent bearing power losses in Eq. (3.3) will be determined here by using one of the empirical formulae available in the literature [24]. This formula defines the bearing mechanical torque loss as M L = ffd 1 β m, (3.4) where d m is the mean bearing diameter, and f 1 is a factor depending on the bearing design and relative bearing load. It is dimensionless, and is determined by y Fs f1 = z, (3.5) C s where the static equivalent load F s and the basic static load rating C s can be found in the manufacturer s catalog while constants y and z depend on the nominal contact angle and the bearing type, and are defined in Ref. [24]. 38

Finally, the loading parameter F β depends on the magnitude and direction of load carried by the bearing. For ball bearings, it is defined as [24] ( ) Fβ = max 0.9Facot α 0.1 Fr, Fr, (3.6) and represents the combined effect of the axial F a and radial F r forces sustained by the bearing. Here, α describes the nominal contact angle, which is 0 for deep-groove ball bearings and 40 for four-point angular contact ball bearings used in this study. The detailed bearing loss calculation parameters of the two types of the bearings used in this study are listed in Table 3.1. For the deep-groove ball bearings, Eq. (3.4) is simplified to ML = ffd 1 r m, (3.7) as they not axially loaded in this set up, as described in Chapter 2. Ball Bearing Type Nominal Contact z y C s ( kn ) dm( mm ) Angle ( ) Radial deep-groove 0 0.0007 0.55 30.5 46 Angular contact 40 0.001 0.33 11.2 46 Table 3.1 Values of parameters for the bearings used in the experiment. 39

With these, the mechanical power loss of a bearing is defined as Pb = MLω, (3.8) and P mesh can be evaluated employing Eq. (3.3). Once the corresponding power losses are found, the total, gearbox mechanical and gear mesh efficiencies can be calculated, respectively, as T P 1 T Pc η =, (3.9a) mech P 1 mech Pc η =, (3.9b) mesh P 1 mesh Pc η =, (3.9c) where the input power into the power circulation loop is defined as Pc = ω Tc. 3.3 Gearbox Mechanical Power Loss Results In this section, gearbox mechanical power loss results are presented. The test matrix of Table 2.1 was implemented for this purpose with the operating conditions defined in Table 2.3 and an automotive transmission fluid provided in a jet lubrication condition at 90 C. The results will be presented in this section in terms of effect of operating conditions (speed and load) as well as effects of basic design parameters (pressure angle, helix angle and module) on the power loss of a gear pair. 40

3.3.1 Influence of Torque and Speed The influences of rotational speed Ω and input torque T c (applied to the loop) on the gearbox mechanical power loss and efficiency are first examined using a 23-tooth (module m = 3.45 ) gear set with Ψ= 30 and Φ= 25.9 (gear set B in Table 2.1 and 2.2), and the 40-tooth (module m = 2.14 ) gear set with Ψ= 21 and Φ= 16.5 (gear set I in Table 2.1 and 2.2), both with ground surfaces (roughness values specified in Table 2.4). Three speed levels of Ω= 2000, 4000 and 6000 rpm (corresponding to the gear pitch line velocities of 16, 32 and 48 m/s, respectively), and four torque levels of T c = 140, 239, 413 and 546 Nm are applied. In Figures 3.1 and 3.2, the gearbox mechanical power loss P mech and mechanical efficiency ηmech are plotted against Ω at different T c levels for the 23-tooth and 40- tooth gears, respectively. It is seen for both designs that P mech increases almost linearly with Ω, with efficiency improving slightly with increased Ω. For instance, at T c = 413 Nm in Figure 3.1(b), η mech = 99.18% at 2000 rpm while η mech = 99.41% at 6000 rpm. This improvement in efficiency can be attributed directly to the fact that the asperity interactions are reduced with increased Ω since the lubricant film thickness is increased. The same sets of data are displayed in Figure 3.3 and Figure 3.4, now in the format of P mech versus T c for the same 23-tooth and 40-tooth gears. Here, measured P mech exhibits significant increases beyond a linear relationship. For instance, at a rotational speed of Ω= 6,000 rpm, the gearbox with 23-tooth gears had a mechanical 41

P mech [ kw ] 3 2 1 140 Nm 239 Nm 413 Nm 546 Nm (a) η mech [%] 0 100.0% 99.5% 2000 4000 6000 140 Nm 239 Nm 413 Nm 546 Nm (b) 99.0% 2000 4000 6000 Ω [rpm] Figure 3.1 Measured (a) P mech and (b) η mech values of the 23-tooth gear pair B having Ψ = 30 and Φ = 25.9 at various torque values. 42

P mech [ kw ] 3 2 1 140 Nm 239 Nm 413 Nm 546 Nm (a) η mech [%] 0 100.0% 99.5% 2000 4000 6000 140 Nm 239 Nm 413 Nm 546 Nm (b) 99.0% 2000 4000 6000 Ω [rpm] Figure 3.2 Measured (a) P mech and (b) η mech values of the 40-tooth gear pair I having Ψ = 21 and Φ = 16.5 at various torque values. 43

3 2 2000 rpm 4000 rpm 6000 rpm (a) P mech [ kw ] 1 0 100.0% 0 100 200 300 400 500 600 2000 rpm (b) 4000 rpm 6000 rpm η mech [%] 99.5% 99.0% 0 100 200 300 400 500 600 Τ c Nm] Figure 3.3 Measured (a) P mech and (b) η mech values of the 23-tooth gear pair B having Ψ = 30 and Φ = 25.9 at various speed values. 44

3 2 2000 rpm 4000 rpm 6000 rpm (a) P mech [ kw ] 1 0 100.0% 0 100 200 300 400 500 600 2000 rpm (b) 4000 rpm 6000 rpm η mech [%] 99.5% 99.0% 0 100 200 300 400 500 600 Τ c Nm] Figure 3.4 Measured (a) P mech and (b) η mech values of the 40-tooth gear pair I having Ψ = 21 and Φ = 16.5 at various speed values. 45

efficiency of η mech = 99.66% at T c = 140 Nm, while its value of η mech = 99.31% at T c = 546 Nm. This is can be viewed as a significant difference, not only in percent mechanical efficiency, but also in terms of net increase in gearbox mechanical power loss P mech. An expected increase in metal-to-metal contact activity with increased T c can be pointed to one of the reasons for this as well. Similar behavior was observed with the other seven gear sets tested in this study as well. 3.3.2 Influence of Helix and Pressure Angles The gearbox mechanical power losses of the 23-tooth gear sets A ( Ψ= 0 and Φ=25.9 ), B ( Ψ= 30 and Φ= 25.9 ) and C ( Ψ= 30 and Φ= 22.7 ) are compared at different speed and torque levels in Figure 3.5 to provide the first comparisons on the influence of Ψ and Φ on P mech. The following observations can be made from Figure 3.5: The gear set C with the lowest normal pressure angle has the highest P mech values except the very lightly loaded conditions. The helical gear set B with the higher Φ value has typically the lowest P mech values except the highest T c value tested. The spur gear power losses improve gradually with increasing T c relative to other two gears sets. It has the highest P mech at T c = 140Nm and the lowest P mech at T c = 546 Nm. 46

1 23T, Hel 0, Pr25.9 Φ= 25.9, Ψ= 0 (A) 23T, Hel30, Pr25.9 Φ= 25.9, Ψ= 30 (B) 23T, Φ= Hel30, 22.7, Ψ= Pr22.7 30 (C) (a) P mech [ kw ] 0.5 0 1 (b) P mech [ kw ] 0.5 0 2000 4000 6000 Ω [ rpm] Figure 3.5 Measured P mech value of the 23-tooth gear sets A to C at (a) T c = 140 Nm, (b) T c = 239 Nm, (c) T c = 413 Nm, and (d) 546 Nm. (Continued) 47

Figure 3.5 Continued, 2 23T, Φ= Hel 25.90,, Pr25.9 Ψ= 0 (A) 23T, Φ= Hel30, 25.9, Ψ= Pr25.9 30 (B) 23T, Φ= Hel30, 22.7, Ψ= Pr22.7 30 (C) (c) P mech [ kw ] 1 0 3 (d) P mech [ kw ] 2 1 0 2000 4000 6000 Ω [ rpm] 48

100.0 23T, Hel 0, Pr25.9 Φ= 25.9, Ψ= 0 (A) 23T, Hel30, Pr25.9 Φ= 25.9, Ψ= 30 (B) 23T, Φ= Hel30, 22.7, Ψ= Pr22.7 30 (C) (a) η mech [%] 99.5 99.0 100.0 (b) η mech [%] 99.5 99.0 2000 4000 6000 Ω [ rpm] Figure 3.6 Measured η mech value of the 23-tooth gears A to C at (a) T c = 140 Nm, (b) T c = 239 Nm, (c) T c = 413 Nm, and (d) T c = 546 Nm. (Continued) 49

Figure 3.6 Continued. 100.0 23T, Hel 0, Pr25.9 Φ= 25.9, Ψ= 0 (A) 23T, Hel30, Pr25.9 Φ= 25.9, Ψ= 30 (B) 23T, Φ= Hel30, 22.7, Ψ= Pr22.7 30 (C) (c) η mech [%] 99.5 99.0 100.0 (d) η mech [%] 99.0 98.0 2000 4000 6000 Ω [ rpm] 50

Among the two helical gear sets (B and C) having the same Ψ and but different Φ, the gear set B with higher Φ has a lower P mech at every T c and Ω value considered. The corresponding η mech values presented in Figure 3.6 confirm the same observations. The helical gear set C has an efficiency value that is 0.15 to 0.05% lower than the other helical gear set B with higher Φ. Overall, gearbox efficiencies vary between 98.99% and 99.66% to reflect the combined influence of T c, Ω, Φ and Ψ. In Figure 3.7, the P mech values measured from all six 40-tooth gear pairs (D to I in Table 2.1) are compared for all T c and Ω values considered. As Figure 3.5 for the 23- tooth gear sets indicate a greater influence of Φ on P mech, the results for these 40-tooth gear pairs are presented in the order of their Φ values, with the gear set I having the lowest Φ= 16.5 as the first gear set and the gear set D having the highest Φ= 26.5 as the last gear set. It is observed in this figure that, among the helical gears, the gear set I with the lowest Φ is one of the least efficient helical gears in this group of five, while the opposite seems to be true for gear set E with the highest Φ. The only spur gear in this figure with a pressure angle of Φ= 26.5 is one of the gear sets with somewhat higher Pmech values, especially at lower toque values and higher speed values. The η mech values corresponding to Figure 3.7 are presented in Figure 3.8. Here the helical gear set E with the highest pressure angle has the one of the highest 51

0.50 40T, 40T, Hel Hel 21, 21, Pr16.54 4 Φ= 16.5, Ψ= 21 (I) Φ= 17.840T,, Ψ= Hel 2020, (H) Pr17.75 40T, Hel Φ= 17.5, Ψ= Pr20.22 4 40T, Hel 17.5, Pr20.22 20.2, 17.5 (G) Φ= 23.440T,, Ψ= Hel30 (F) (a) P mech [ kw ] 0.25 40T, 40T, Hel30, Hel30, Pr26.5 Pr26.5 4 Φ= 26.5, Ψ= 30 (E) Φ= 26.540T,, Ψ= Hel 0 0,Pr2 (D) 0.00 1.00 40T, Hel 21, Pr16.54 40T, Hel 20, Pr17.75 40T, Hel 17.5, Pr20.22 40T, Hel30, Pr24.7 40T, Hel30, Pr26.5 40T, Hel 0, Pr28 (b) P mech [ kw ] 0.50 0.00 2000 4000 6000 Ω [ rpm] Figure 3.7 Measured P mech value of the 40-tooth gears D to I at (a) T c = 140 Nm, (b) T c = 239 Nm, (c) T c = 413 Nm, and (d) T c = 546 Nm. 52 (Continued)

Figure 3.7 Continued. 2.00 40T, 40T, Hel Hel 21, 21, Pr16.54 4 Φ= 16.5, Ψ= 21 (I) Φ= 17.8 40T,, Ψ= Hel 20 20, (H) Pr17.75 40T, Hel Φ= 17.5, Ψ= Pr20.22 4 40T, 20.2 Hel, 17.5, 17.5 Pr20.22 (G) Φ= 23.40T,, Ψ= Hel30 (F) (c) P mech [ kw ] 1.00 40T, 40T, Hel30, Hel30, Pr26.5 4 Φ= 26.5, Ψ= 30 (E) Φ= 26.5 40T,, Ψ= Hel 00, (D) 0.00 3.00 (d) 2.00 P mech [ kw ] 1.00 0.00 2000 4000 6000 Ω [ rpm] 53

η mech [%] 100.00 99.75 40T, Hel 21, 40T, Hel 20, Φ= 17.8, Ψ= Pr17.75 Φ= 16.5, Ψ= 21 (I) 20 (H) Φ= 20.2, Ψ= 17.5 (G) 40T, Hel 17.5, 40T, Hel30, Φ= 23.4, Ψ= 30 (F) 40T, Hel30, Pr26.5 Φ= 26.5, Ψ= 30 (E) 40T, Hel 0, Φ= 26.5, Ψ= 0 (D) (a) 99.50 100.0 40T, Hel 21, Pr16.54 40T, Hel 20, Pr17.75 40T, Hel 17.5, 40T, Hel30, Pr24.7 (b) η mech [%] 99.5 40T, Hel30, Pr26.5 40T, Hel 0, 99.0 2000 4000 6000 Ω [ rpm] Figure 3.8 Measured η mech value of the 40-tooth gears D to I at (a) c140 Nm, (b) T c = 239 Nm, (c) T c = 413 Nm, and (d) T c = 546 Nm. 54 (Continued)

Figure 3.8 Continued. 100.0 40T, Hel 21, Pr16.54 (I) 40T, Hel 17.820, Φ=, Ψ= Pr17.75 Φ= 16.5, Ψ= 21 20 (H) 40T, Φ= 20.2 Hel 17.5,, Ψ= 17.5 (G) 40T, Φ= Hel30, Pr24.7 Ψ= 23.4, 30 (F) (c) η mech [%] 99.5 40T, Hel30, Pr26.5 Φ= 26.5, Ψ= 30 (E) 40T, Hel 0, Φ= 26.5, Ψ= 0 (D) 99.0 100.0 40T, Hel 21, Pr16.54 40T, Hel 20, Pr17.75 40T, Hel 17.5, 40T, Hel30, Pr24.7 (d) η mech [%] 99.5 40T, Hel30, Pr26.5 40T, Hel 0, 99.0 2000 4000 6000 Ω [ rpm] 55

efficiencies. The spur gear pair D is also one of the more efficient gear sets in the populations at higher torque values (Figure 3.8(c) and (d)). Yet, the range of η mech values is rather narrow when the value of T c is high. In Figure 3.8(b), for instance, with T c = 239 Nm and Ω= 6, 000 rpm, the spread between the lowest (gear set I with Φ=16.5 and Ψ= 21 ) and the highest (gear set E with Φ= 26.5 and Ψ= 30 ) η mech values is only 0.15%, suggesting that the influences of neither Φ nor Ψ on η mech is that dominant while Φ can be said to be more influential than Ψ. Meanwhile, a comparison of Figure 3.6 and 3.8 indicate that the normal module of the gears has a more prominent impact on gearbox mechanical efficiency η mech. For instance, the gear set B (23-tooth, Φ= 25.9 and Ψ= 30 ) in Figure 3.6 at T c = 546 Nm and Ω= 6, 000 rpm has η mech = 99.31%, confirming earlier conclusions of Xu et al. [12] and Petry-Johnson et al. [23] in regards to the influence of module on gear efficiency. 3.4 Gearbox Spin Power Loss Results The load independent spin power loss Pspin was measured for each test specified in Table 2.1 by setting T c = 0 Nm (i.e. applying very little torque at the split coupling to maintain contact). The measured P spin values are compared in Figure 3.9 for all nine gear sets tested. 56

0.50 23T, Hel 0, 25.9 Pr25.9 Φ=, Ψ= 0 (A) (a) P spin [ kw ] 0.25 23T, Hel30, 25.9, Pr25.9 Φ= Ψ= 30 (B) (C) 23T, Φ= 22.7 Hel30,, Ψ= Pr22.7 30 0.00 0.50 P spin [ kw ] 0.25 40T, Hel 21, Pr16.54 40T, 4 Hel 17.820, Φ=, Ψ= Pr17.75 Φ= 16.5, Ψ= 21 (I) 20 (H) 40T, Hel 17.5, Pr20.22 (G) 40T, Hel30 40T, Hel 17.5, Pr20.22 4 Φ= 20.2, Ψ= 17.5 Φ= 23.4, Ψ= 30 (F) 40T, Hel30, Pr26.5 40T, Φ= 4 Hel 26.50,Pr2 Φ= 26.5, Ψ= 30 (E), Ψ= 0 (D) (b) 0.00 2000 4000 6000 Ω [ rpm] Figure 3.9 Measured P spin values of (a) 23-tooth gear sets A to C and (b) 40-tooth gear sets D to I. 57

As expected, the measured Pspin values are very close to each other (within 0.05 kw) suggesting that neither pressure angle nor the helix angle is a primary factor impacting P spin, confirming results of Seetharaman and Kahraman [25]. The exponential relationship between the rotational speed and P spin is also evident in this figure. 3.5 Conclusion In this chapter, the power loss and efficiency results corresponding to the experimental test matrix and test specimens presented in Chapter 2 have been presented. The combined influence of T c, Ω, Φ and Ψ on 23-tooth and 40-tooth gear sets have been demonstrated. These results indicate that the gear module and pressure angle are more significant than the helix angle in terms of their influence on gearbox power losses. These experimental results will be used in Chapter 4 to validate an efficiency model that was developed in a companion study. The validated model will then be used in Chapter 5 to better quantify the influence of various gear design parameters on the mechanical efficiency of helical gear sets. 58

CHAPTER 4 COMPARISON OF EXPERIMENTAL RESULTS TO PREDICTIONS OF A GEAR EFFICIENCY MODEL 4.1 Introduction In this chapter, the experimental results of gear efficiency tests on spur and helical gears performed under jet-lubricated conditions are compared with the predictions of a gear efficiency model first developed by Xu et al. [12] and enhanced later by by Li et al. [1] through addition of a new friction model. The experimental gearbox mechanical power losses P mech were processed further according to Eq. (3.3) by removing bearing losses using simplified formulae from Ref. [24] to obtain gear mesh mechanical power losses P mesh. The model is described briefly in the next section focusing mostly on the new friction coefficient model, followed by a section of comparisons between the predicted and measured Pmesh values. Based on these comparisons, the accuracy of the helical gear pair efficiency model is discussed at the end. 59

4.2 Gear Mesh Mechanical Power Loss Model The gear mesh mechanical power loss model used in this study was first developed by Xu et al. [12]. Referring to Xu et al. [12] for details of this model, a brief description of its methodology will be given here. This model uses the instantaneous friction coefficient µ (, z θφ, m) at each contact point (, z θ ) at each rotational angle φ m ( m = 1 to M) to compute the friction force at each contact position as Ff (, z θφ, m) = µ (, z θφ, m) W(, z θφ, m) (4.1) where W(, z θφ, m) is the normal load at the contact point of interest. The instantaneous mechanical power loss and the mechanical efficiency of the gear are then found as Q Pmesh ( m) = [ (, z, m) W(, z, m) us (, z, m) ] q q= 1 φ µ θφ θφ θφ, (4.2a) P ( ) ( ) 1 mesh m mesh m = φ η φ Tinωin (4.2b) where Q is the total number of loaded contact segments at a given mesh position φ m, T in is the input torque, us = u1 u2 is the sliding velocity, and ω in is the input speed. These instantaneous values are averaged over an entire mesh cycle to obtain the average gear mesh mechanical power loss and mechanical efficiency as M P 1 mesh Pmesh ( m) M m = 1 = φ, (4.3a) 60

η mesh P = 1 mesh Tin in ω. (4.3b) In Eq. (4.1), the normal load W(, z θφ, m) at a contact point in the gear mesh interface can be predicted conveniently by using the Load Distribution Model (LDP). Determining the friction coefficient µ (, z θφ, m) at the same point, however, is a more complex task as it must involve elastohydrodynamic lubrication (EHL) analysis. Xu et al. [12] relied on a previous EHL model by Cioc et al. [15] for this task. Since each EHL analysis takes a significant amount of computational time and one must perform hundreds of such analyses to predict µ (, z θφ, m) at every contact point of interest at every rotational increment, Xu et al. [12] focused their methodology on development of a closed-form friction formula based on regression analysis of a large number of EHL runs. They defined ranges for (i) rolling velocity, (ii) sliding velocity, (iii) radii of curvature, (iv) normal load, (v) lubricant temperature (or viscosity) and (vi) surface roughness amplitudes representative of automotive gear applications and ran several thousands of EHL analyses using the EHL model of Cioc et al. [15] with a typical gear oil (75W90). They applied a regression analysis to the predicted µ values to obtain a single formula to determine the friction coefficient as a function of the contact parameters listed above. 4.2.1 A Mixed EHL Model to Predict the Friction Coefficient While the model of Xu et al. [12] was reasonably accurate in predicting mechanical efficiency of high speed gear pairs such as those tested by Petry-Johnson et al. [23], it had difficulties in handling conditions where the lubrication conditions are not 61

optimal such that actual asperity contacts are of common occurrence. The root cause of this shortcoming was the EHL model of Cioc et al. [15], which could handle only a few asperity contacts without any numerical difficulties. In order to remedy this, Li et al. used a robust mixed EHL model [17] to derive a new µ formula where the asperity interactions of any level are included accurately. While the mixed EHL model of Li and Kahraman [17] was a 2D model for any generic point contact problem, a 1D version of it will be outlined here for line contact problems such as those in gear contacts [1]. For any contact point in the gear mesh interface the transient Reynolds equation governs the non-newtonian fluid flow in the contact areas with no asperity interactions [17] p ( ur ρ h) ( ρ h) fx = +, x x x t (4.4) where the parameters p, h and ρ denote the pressure, thickness and density of the fluid, respectively, all of which are dependent on x and t, and u 1 r = 2 ( u1+ u2 ) is the rolling velocity. The fluid flow coefficient is defined as [26] fx 3 ρh τ cosh( m ), 12η τ0 = (4.5) where η is the lubricant viscosity, τ 0 is any lubricant reference stress, and the viscous shear stress 1 m = 0sinh [ us( t) ( 0h)]. Here, us = u1 u2 is sliding velocity of the τ τ η τ contact. Equation (4.1) describes the lubricant flow within the contact regions where the 62

fluid film thickness is greater than zero such that the surfaces are separated. In the regions where the asperity contact occurs ( h = 0 ), the Reynolds equation is set to [17] ( ur ρ h) ( ρ h) + = 0. x t (4.6) Assuming a smooth transition between the fluid and asperity contact areas, Eq. (4.4) and (4.6) form a unified Reynolds equation system that governs the mixed EHL behavior of the contact, considering both the fluid and the asperity contact regions simultaneously. The film thickness of a contact point at any coordinate x at time t under elastic conditions is defined as [1, 17] hxt (,) = h0() t + g0() x + V(,) xt R1(,) xt R2(,) xt (4.7) where h 0 () t is the reference film thickness, and R 1 (,) xt and R 2 (,) xt are the roughness profiles of the two surfaces at time t. Here R 1 (,) xt and R 2 (,) xt move at velocities u 1 and u 2 that correspond to slide-to-roll ratio SR = us ur = 2( u1 u2) ( u1+ u2). The term g 0 ( x ) is the geometric unloaded gap between the two surfaces that is given as a function of the equivalent radius of curvature [ ] req () t = r1() t r2() t r1() t + r2() t and the x coordinate in the direction of sliding as 2 g0 ( x ) = x (2 r eq ). 63

Additionally, the elastic deformation V(,) xt due to the applied normal load W is given as [1, 17] xe V (,) x t = K( x x ) p( x,) t dx, (4.8) x s In the above equation, x s and x e are the start and end points of the computational domain of the contact zone. K( x) = 4 ln x ( π E ) is the so-called influence coefficient where 2 2 1 υ2 1 υ 1 E = 2 + E E 1 2 1 with υ i and E i are the Poisson s ratio and the Young s modulus of contact body i. Li and Kahraman [17] used a two-slope viscosity-pressure model [26] and a density-pressure relationship [13] along with Eq. (4.4) to (4.8) to predict the pressure and film thickness distributions, pxt (,) and hxt (,). A load balance equation W = p(,) x t dx (4.9) L was used to check whether the total contact force due to the pressure distribution (both hydrodynamic and asperity contact) over the entire contact area is balanced by the normal tooth load applied at that instant. The reference film thickness h 0 () t in Eq. (4.7) is adjusted within a load iteration loop until Eq. (4.9) is satisfied. 64

4.2.2 Derivation of a New Friction Coefficient Formula for Gear Contacts Li et al. [1] defined the viscous shear stress acting on the surface of gear 1 as h p * ( u2 u1) q = + ηx, 2 x h (4.10) * where ηx = η cosh( τm τ0) is the effective viscosity in the direction of rolling. Within the asperity interaction regions, the shear stress was defined as [1] q= µ d p (4.11) In Eq. 4.11, µ d is the coefficient of friction for dry contact condition. With these, the friction coefficient at a given time instant t was computed by xe µ = q(,) x t dx. (4.12) x s Each simulation is carried out for a total of N time steps ( N = 1000 in Ref [1]) and the average friction coefficient for the contact condition considered was defined as µ 1 = µ t. N t Nt n= 1 n As it was done originally by Xu et al. [12], instead of using the mixed EHL model directly in real time to determine the traction force at every contact cell on the tooth surface, an upfront parametric study was performed by considering all combinations of the key parameter values within the ranges defined in Table 4.1, covering most of the 65

intended automotive applications with a typical automatic transmission fluid. In addition, a measured roughness profile from a shaved gear surface with R q = 0.5μm was considered and profiles with different R a values were obtained from this baseline profile by multiplying it by a constant. It was assumed for practical purposes that a contact with Rq 1 = a and Rq 2 = b is equivalent to a surface with Rq 1 = b and shows the mechanical properties of the automatic transmission fluid used here. Rq 2 = a. Table 4.2 A total of 42,000 combinations covering a wide range of contact conditions experienced by automotive gearing are obtained from the parameter values listed in Table 4.1. In order to illustrate this, the µ values predicted by the mixed EHL model for these combinations are plotted against the lambda ratio λ (ratio of the minimum film thickness calculated by using Dowson-Higginson formula for line contacts [13] to the RMS surface roughness value) in Figure 4.1. Several observations can be made from Figure 4.1. First of all, contact conditions vary significantly, ranging from boundary EHL conditions (with λ values as small as 0.02, indicating very undesirable asperity contact activity) to full film micro EHL conditions (very thick lubricant films having λ as large as 30 with no asperity contacts at all). More importantly, the µ versus λ plot shows a piecewise linear relationship. For λ 1, µ varies quite linearly with λ with a very steep negative slope representing the boundary or mixed EHL conditions, while another linear relation with much smaller negative slope is observed for λ > 1 when there is little or no asperity interaction. 66

Lubricant Hertzian Pressure A transmission fluid p h [GPa] 0.5, 1, 1.5, 2, 2.5 Equivalent Radius of Curvature r eq [mm] 5, 20, 40, 80 Rolling Velocity u r [m/s] 1, 5, 10, 15, 20 Slide to Roll Ratio SR 0.025, 0.05, 0.1, 0.25, 0.5, 0.75, 1 Inlet Lubricant Temp. T [ C] 25, 50, 75, 100 Surface 1 roughness RMS Surface 2 roughness RMS R q [μm] 1 0.1, 0.2, 0.4, 0.6, 0.8 R q [μm] 2 0.1, 0.2, 0.4, 0.6, 0.8 Table 4.1 Parametric design for the development of a friction coefficient formula 67

Temperature T [ C] Pressure-viscosity Coefficient α -1 1 [GPa ] Dynamic Viscosity η 0 [Pa s] Density ρ 3 0 [kg/m ] 25 18.7 0.04786 864.80 50 15.6 0.01781 849.14 75 13.5 0.00858 833.48 100 12.0 0.00490 817.82 Table 4.2 Basic parameters of the automatic transmission fluid used in this study 68

The EHL µ data shown in Figure 4.1 was divided into two subsets at the threshold value of λ = 1, and a regression analysis is carried out for each subset separately [1]. The first order parameters selected for the general linear regression model for the response parameter of ln µ were λ, SR, W = log 10( ph E ), α = log 10( α 1E ), req log 10( req E W ) =, V log [ u η W ] =, and two roughness parameters 10 r 0 Sc = SE c W and Seq = Seq E W. Here, W = W L was the load density along the contact line of length L, 2 2 c = + q q and S R R 1 2 S = R R ( R + R ). In addition, eq q1 q2 q1 q2 the higher order parameters, which are the combination of those first order parameters, are also considered. Utilizing the statistical software, MINITAB, the following regressed µ formula was obtained with an adjusted R-square value of about 92% [1] For λ 1: W µ = exp a 0 + a1λ S + awr 2 + a3λ + a4s a a 5λ a6λ 7( SR) + a8v + a9ln λ ( SR) S S, eq λ c eq eq c (4.13a) For λ > 1: req W = exp b0 + ( bv 1 + b2req ) + bv 3 + b4 + b5 µ λ α α ( ) ( ) req V b7+ bv 8 b9 + b10 ln λ+ b11α b12λ b b 13λ + 6r eq λ SR W Sc. (4.13b) Here, coefficients a 0 to a 9 and b 0 to b 13 are constants representative of the lubricant considered. 69

0.3 0.25 µ 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 30 35 Lambda Ratio λ Figure 4.1 The relationship between the friction coefficient µ and lambda ratio λ. 70

4.3 Comparison of Measured and Predicted Power Loss Values The helical gear mechanical efficiency tests presented in Chapter 2 are simulated here using the model described in the previous section, and the predictions for each of the test gear pairs are compared to those measurements presented in Chapter 3. Here, in order to separate the measured gear mesh power loss P mesh from the gearbox power loss, the bearing power loss formula given in Sect. 3.2 was used. The predicted and measured values for 23 tooth gear sets A, B and C are shown in Figure 4.2, 4.3 and 4.4, respectively. In these figures, differences in P mesh between the experiments and predictions for all three gear sets at speeds Ω 4000 rpm and toque values Tc 413 Nm are all less than 0.1 kw. Differences are slightly larger at the highest speed and load conditions, which might be due to the simplicity of the bearing power loss model used. In general, agreement between the predictions and the measurements is rather good. The predicted and measured values of gear mesh mechanical power loss for 40 tooth gear sets D through I are shown in Figures 4.5 through 4.10. The net differences at lower values of speed and torque are minimal, and those at higher values of speed and torque are slightly larger, but for all the gears, the predictions follow the same trend as the experiments. The maximum difference between the predicted and measured values of power loss is less than 0.1 kw for both the 23 and 40-tooth gear pairs at 140 Nm and 239 Nm. At higher values of load, 413 Nm and 546 Nm, the absolute differences are higher in some cases, amounting to maximum percentage differences that are less than 0.2%. 71

1.5 Prediction Measurement (a) 1.0 P mesh [ kw ] 0.5 0.0 1.5 0 200 400 600 Prediction (b) Measurement 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.2 Comparison of predicted and measured P mesh values of 23-tooth gear pair A (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm. 72 (Continued)

Figure 4.2 Continued. 2.0 1.5 Prediction Measurement (c) P mesh [ kw ] 1.0 0.5 0.0 0 200 400 600 Tc [ Nm] 73

1.0 Prediction Measurement (a) P mesh [ kw ] 0.5 0.0 1.5 0 200 400 600 Prediction (b) Measurement 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.3 Comparison of predicted and measured P mesh values of 23-tooth gear set B (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000rpm. (Continued) 74

Figure 4.3 Continued. 2.0 1.5 Prediction Measurement (c) P mesh [ kw ] 1.0 0.5 0.0 0 200 400 600 Tc [ Nm] 75

1.5 Prediction Measurement (a) 1.0 P mesh [ kw ] 0.5 0.0 2.0 1.5 0 200 400 600 Prediction (b) Measurement P mesh [ kw ] 1.0 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.4 Comparison of predicted and measured P mesh values of 23-tooth gear pair C (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm. 76 (Continued)

Figure 4.4 Continued. 3.0 Prediction Measurement (c) 2.0 P mesh [ kw ] 1.0 0.0 0 200 400 600 Tc [ Nm] 77

1.0 Prediction Measurement (a) P mesh [ kw ] 0.5 0.0 1.5 0 200 400 600 Prediction (b) Measurement 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.5 Comparison of predicted and measured P mesh values of 40-tooth gear pair D (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm. 78 (Continued)

Figure 4.5 Continued. 1.5 Prediction Measurement (c) 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] 79

1.0 Prediction Measurement (a) P mesh [ kw ] 0.5 0.0 1.0 0 200 400 600 Prediction (b) Measurement P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.6 Comparison of predicted and measured P mesh values of 40-tooth gear set E (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm. 80 (Continued)

Figure 4.6 Continued. 1.5 Prediction Measurement (c) 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] 81

1.0 Prediction Measurement (a) P mesh [ kw ] 0.5 0.0 1.5 0 200 400 600 Prediction (b) Measurement 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.7 Comparison of predicted and measured P mesh values of 40-tooth gear set F (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm. 82 (Continued)

Figure 4.7 Continued. 2.0 Prediction Measurement (c) P mesh [ kw ] 1.0 0.0 0 200 400 600 Tc [ Nm] 83

1.0 Prediction Measurement (a) P mesh [ kw ] 0.5 0.0 1.0 0 200 400 600 Prediction (b) Measurement P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.8 Comparison of predicted and measured P mesh values of 40-tooth gear set G (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm. 84 (Continued)

Figure 4.8 Continued. 1.5 Prediction Measurement (c) 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] 85

1.0 Prediction Measurement (a) P mesh [ kw ] 0.5 0.0 1.5 0 200 400 600 Prediction (b) Measurement 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.9 Comparison of predicted and measured P mesh values of 40-tooth gear set H (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm. 86 (Continued)

Figure 4.9 Continued. 2.0 Prediction Measurement (c) P mesh [ kw ] 1.0 0.0 0 200 400 600 Tc [ Nm] 87

1.0 Prediction Measurement (a) P mesh [ kw ] 0.5 0.0 1.5 0 200 400 600 Prediction (b) Measurement 1.0 P mesh [ kw ] 0.5 0.0 0 200 400 600 Tc [ Nm] Figure 4.10 Comparison of predicted and measured P mesh values of 40-tooth gear set I (Table 2.1) at (a) 2000 rpm, (b) 4000 rpm, and (c) 6000 rpm. 88 (Continued)

Figure 4.10 Continued. 2.0 Prediction Measurement (c) P mesh [ kw ] 1.0 0.0 0 200 400 600 Tc [ Nm] 89

Next the same measured and predicted P mesh values are presented as a function of key gear or gear pair parameters to (i) demonstrate that the model is capable of predicting the measured sensitivities, and (ii) to identify the key parameters influencing P mesh. As the first sensitivity of these, Figure 4.11 shows the predicted and measured P mesh values against the normal pressure angle Φ values of the nine test gear pairs. In this figure, the first column shows the predicted P mesh at two torque values of T c = 140 Nm and 546 Nm (both at Ω= 2000 rpm). The second column is formed by the measured P mesh values for the same. Viewing 23-tooth gears (solid symbols) and 40- tooth gears separately, it is seen that an increase in Φ reduces P mesh. The predicted and measured sensitivities of P mesh to Φ are in good agreement. The gap between the 23- tooth and 40-tooth gear sets is due to the influence of the module that will be demonstrated later. Figure 4.12 uses the same format and conditions as Figure 4.11 to show the predicted and measured P mesh values against the helix angle Ψ values of the same nine test gear pairs. Here, the influence Ψ on both predicted and measured P mesh values is very slight, suggesting that increasing Ψ may not be an effective way of increasing efficiency of the gear pair. In Figure 4.13, the same data is presented, now with normal module m as the parameter. In this case, a significant increase in P mesh values is observed both experimentally and theoretically when m is increased. 90

0.4 A B C D E F G H I (a1) (a2) 0.2 P mesh [ kw ] 0.0 1.2 (b1) (b2) 0.8 0.4 0.0 15 20 25 30 15 20 25 30 Φ [deg.] Figure 4.11 Variation of P mesh of test gear pairs with normal pressure angle Φ at Ω= 2000 rpm and (a) T = 140 Nm and (b) T = 546 Nm. (a1, b1) c predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs. c 91

0.4 A B C D E F G H I (a1) (a2) 0.2 P mesh [ kw ] 0.0 1.2 (b1) (b2) 0.8 0.4 0.0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Ψ [deg.] Figure 4.12 Variation of P mesh of test gear pairs with helix angle Ψ at Ω= 2000 rpm and (a) T c = 140 Nm and (b) T c = 546 Nm. (a1, b1) predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs. 92

0.4 A B C D E F G H I (a1) (a2) 0.2 P mesh [ kw ] 0.0 1.2 (b1) (b2) 0.8 0.4 0.0 0.5 1.5 2.5 3.5 4.50.5 1.5 2.5 3.5 4.5 m [ mm] Figure 4.13 Variation of P mesh of test gear pairs with module m at Ω= 2000 rpm and (a) T c = 140 Nm and (b) T c = 546 Nm. (a1, b1) predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs. 93

0.4 A B C D E F G H I (a1) (a2) 0.2 P mesh [ kw ] 0.0 1.2 (b1) (b2) 0.8 0.4 0.0 1.0 1.5 2.01.0 1.5 2.0 ε p Figure 4.14 Variation of P mesh of test gear pairs with profile contact ratio ε p at Ω= 2000 rpm and (a) T = 140 Nm and (b) T = 546 Nm. (a1, b1) c predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs. c 94

0.4 A B C D E F G H I (a1) (a2) 0.2 P mesh [ kw ] 0.0 1.2 (b1) (b2) 0.8 0.4 0.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 ε f Figure 4.15 Variation of P mesh of test gear pairs with face contact ratio ε f at Ω= 2000 rpm and (a) T = 140 Nm and (b) T = 546 Nm. (a1, b1) c predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs. c 95

0.4 A B C D E F G H I (a1) (a2) 0.2 P mesh [ kw ] 0.0 1.2 (b1) (b2) 0.8 0.4 0.0 1.0 2.0 3.0 4.0 1.0 2.0 3.0 4.0 ε t Figure 4.16 Variation of P mesh of test gear pairs with total contact ratio ε t at Ω= 2000 rpm and (a) T = 140 Nm and (b) T = 546 Nm. (a1, b1) c predictions, and (a2, b2) measurements. Solid symbols represent 23-tooth gear pairs. c 96

The next three figures, Figure 4.14, 4.15 and 4.16, show the influence of profile ( ε p ), face ( ε f ) and total ( ε t = ε p + ε f ) contact ratios on predicted and measured P mesh values. Increasing ε p appears to increase P mesh while a slight reduction in P mesh is observed with increased ε f to these gear pair parameters agree reasonably well.. The predicted and measured trends in terms of sensitivity 4.4 Conclusions In view of the direct comparisons presented in Figures 4.2 to 4.10 and trends in Figures 4.11 to 4.16, it can be concluded that the model is capable of predicting the mechanical power losses of spur and helical gear pairs of various designs successfully over wide ranges of operating speed and torque. The minor differences between the model and the predictions can be attributed to one or more of the following reasons: (i) inaccuracies in the bearing efficiency model used, (ii) uncertainty in the measured data, and (iii) the prediction errors originating from the regression analysis or the definition of the simulation parameters. Yet, the overall performance of the model can be considered to be rather good. Accordingly, the model can be used for design optimizations studies presented in the next chapter. 97

CHAPTER 5 PARAMETER SENSITIVITY STUDY 5.1 Introduction In this chapter, the gear pair mechanical efficiency model that was introduced in Chapter 4 will be exercised in association with the RMC program of the Ohio State University to provide an example parameter sensitivity study. The results of this study will be used to quantify the impact of key design parameters on gear pair mechanical power loss. They will also be used to demonstrate the challenges in defining a particular gear design that represents the optimal solution in terms of all the functional requirements. Based on this study, a candidate design that balances all critical (noise, durability and efficiency) requirements well will be used to perform a micro-geometry study to show the influence of tooth surface corrections on the same outcome metrics. It is noted here that RMC has been used successfully to balance noise and durability outcome of gear designs. This parametric study adds the efficiency into the process. The predicted parameters in this study will include peak-to-peak transmission error (TE) as a noise metric, maximum contact and root bending stresses ( σ c and σ b ) as 98

gear fatigue life metrics. In addition, the mechanical power loss of the gear pair P mesh is computed using the proposed model to represent the efficiency of the gear design. The ultimate goal here is to find a design that results in minimum values for all P mesh, TE, σ c, and σ b. Parameter Range Increment Number of teeth, N 20-53 1 Helix angle, Ψ [deg] 15-30 1 Normal pressure angle, Φ [deg] 15-30 2.5 Working depth constant, Λ wd 1.85-3.378 0.218 Table 5.1 Values of the gear parameters varied in the parameter sensitivity study. 99

5.2 Analysis Details An example condition with a fixed center distance of 91.5 mm and unity (1:1) ratio was considered here. Both gears of the pair (gear and pinion) were kept identical to reduce the number of parameters. In addition, the face width of all gears was kept constant at 25 mm. Four basic gear design parameters, normal pressure angle ( Φ ), helix angle ( Ψ ), the number of teeth (N) that is the same for both the pinion and gear and the working depth ratio ( Λ wd ) are varied. The working depth constant is defined here as the ratio of the sum of addendum and dedendum of the gears to normal module ( m ). Table 5.1 presents the ranges over which the basic parameters were varied, and the increments for each parameter. Here Λ wd is varied between 1.8 and 3.5 at an increment of 0.218 (8 levels) to obtain a range of tooth heights. Meanwhile, N is varied from 20 to 53 with an increment of 1 to represent a range of fine to coarse gear teeth. Discrete values between 15 and 30 degrees are considered for both Φ (increment of 2.5 deg.) and Ψ (increment of 1 deg.). A profile crown of M P = 15 µm and a circular lead crown of M L = 12 µm are applied to all of the gears. An automatic transmission fluid at 90 C is used in the simulations. Input torque and speed are kept at T in = 250 Nm and Ω= 60 4, 000 rpm. 2π ω = The surface roughness amplitudes of the gears are kept constant at R = R = 0.4 µ m.. q q 1 2 100

1 0.8 P mesh [ kw ] 0.6 0.4 0.2 0 18 28 38 48 58 N 1 2 3 4 5 m [ mm] 1 0.8 P mesh [ kw ] 0.6 0.4 0.2 0 93 95 97 99 101 103 do [ mm] 0.3 0.8 1.3 1.8 2.3 2.8 ε f Figure 5.1 Variation of P mesh with (a) N, (b) m, (c) d o, (d) ε f, (e) ε p, and (f) ε t. (Continued) 101

Figure 5.1 Continued. 1 0.8 P mesh [ kw ] 0.6 0.4 0.2 0 0.8 1.3 1.8 2.3 2.8 1.3 2.3 3.3 4.3 5.3 ε p ε t 102

All combinations of values of all four parameters shown in Table 5.1 resulted in more than 30,000 different designs, of which only 13,500 were deemed acceptable as the rest were eliminated due to reasons such as inadequate backlash, inadequate tip thickness or root clearance (minimum acceptable backlash, tip thickness and root clearance values were chosen as 0.04, 0.2 and 0.2, respectively). 5.3 Variation of Gear Mesh Mechanical Power Loss with Gear Design parameters In Figure 5.1(a-c), the variation of the P mesh of all of the 13,500 designs are plotted against N, m and outside (major) diameter d o, respectively (small lighter color dots). In Figure 5.1(a), about a 0.3 to 0.6 kw spread in P mesh is observed for each N value due to variations coming from the other parameter ranges. In general, P mesh tends to reduce with increase in number of teeth. When the other parameters are fixed at Φ=22.5, Ψ= 20 and Λ wd = 2.286, the database is reduced to 33 designs represented by larger symbols in Figure 5.1(a). This subset of data points shows the sole influence of N on P. Along the line formed by these data points, Pmesh 0.6 kw for N = 20 teeth, which is reduced to half when the number of teeth is increased to 50. This suggests that finer pitch gears should have lower mechanical power losses as observed in the experimental results described in Chapter 3 and as well as other investigators [12, 23]. Figure 5.1(b) that plots P mesh against normal module m supports the same conclusion in an even more convincing way. The 13,500 designs considered cover a module range of 103

1.5 to 4.4 mm. Again focusing on the subset of data with Φ= 22.5, Ψ= 20 and Λ wd = 2.286, an almost linear relationship between P mesh and m is observed in Figure 5.1(b), obtained by changing N. In Figure 5.1(c), meanwhile, variation of P mesh with d o of gears is illustrated. The general trend observed from the larger population is that P mesh increased with d o. In other words, longer teeth result in more sliding, which increases power losses as well. Looking at the same sub-set of the data as Figure 5.1(a, b), P mesh is shown to increase in a linear manner with d o, changing from 0.25 kw at d = 94 mm to 0.6 kw at a value of d = 99 mm. o o The same data points are plotted in Figure 5.1(d-f) to show the variation of P mesh with the theoretical values of the face contact ratio ( ε f ), the profile contact ratio ( ε p ) and the total contact ratio ( ε t = ε f + ε p ). Here, there is no general trend between P mesh and ε f and ε p. This is because contact ratios can be changed in different ways that might cause reductions or increases in P mesh. For instance, ε p can be increased conveniently by reducing module as well as increasing the major diameter of gears. The former was shown in Figure 5.1(b) to reduce the P mesh while the latter in Figure 5.1(c) was shown to increase it. As a result, the 13,500 designs plotted in Figure 5.1(e) do not follow a trend. Meanwhile, the same subset of data with Φ= 22.5, Ψ= 20 and Λ wd = 2.286 (larger symbols) indicate that P mesh reduces with both ε f and ε p, the influence of ε p being more drastic. 104

Figure 5.2(a) plots P mesh versus normal pressure angle Φ, indicating that an increase in Φ helps reduce P mesh. Defining a sub-set of the data with N = 30 teeth, Ψ=20 and Λ wd = 2.286, the same figure also shows the variation of this group of designs with Φ. This subset of data shows that P mesh is reduced from almost 0.8 kw to 0.4 kw by increasing Φ from 15 to 25. The same trend is observed in Figure 5.2(b) for Ψ, but now in a less significant way. On the same figure, it was also shown that increasing Ψ from 15 to 30 for a group of candidate designs with N = 30 teeth, Φ=22.5 and Λ wd = 2.286 reduces P mesh by only about 20%. Finally, in Figure 5.2(c), the sole influence of Λ wd is illustrated, with a sub-set formed by designs having N = 30, Φ= 22.5 and Ψ= 20 showing a steep increase in P mesh with Λ wd. 5.4 Selection of an Optimum Design Satisfying Multiple Performance Criteria Figures 5.1 and 5.2 reveal certain trends that help reduce P mesh such as increasing Φ or reducing d o. Some of these trends can be expected to increase noise excitations while hurting the contact and bending stresses as well. In other words, a design that is very good in terms of power loss characteristics (or efficiency) might indeed be very poor for its other attributes. 105

1 0.8 P mesh [ kw ] 0.6 0.4 0.2 0 14 17 20 23 26 29 32 Φ [deg.] 1 0.8 P mesh [ kw ] 0.6 0.4 0.2 0 14 17 20 23 26 29 32 Ψ [deg.] 1 0.8 P mesh [ kw ] 0.6 0.4 0.2 0 1.5 2.0 2.5 3.0 3.5 Λ wd Figure 5.2 Variation of P mesh with (a) Φ, (b) Ψ, and (c) Λ wd. 106

In order to illustrate this, the same 13,500 designs were evaluated for their peakto-peak transmission error (TE), maximum contact stress and maximum root bending stress ( σ c and σ b ). Figure 5.3(a-c) shows the variation of P mesh with TE, σ c, and σ b, respectively. In Figure 5.3(a), P mesh versus TE, a candidate Design A (with parameters listed in Table 5.2) is chosen that has the lowest P mesh and TE values (i.e. most efficient and potentially the quietest design). The same Design A is also located in Figure 5.3(b, c) to show its corresponding σ c and marginal in terms of bending strength as it has a very high σ b values. It is clear here that Design A is rather σ c and it is absolutely one of the worst designs in terms of its σ b value. Similarly, in Figure 5.3(c), another design, Design B, is chosen based on combined P mesh and σ b as a favorable design and located in Figure 5.3(a, b) to show that it has very high TE value. Similarly, Design C that is good for P mesh and σ c in Figure 5.3(b) is shown to have very high σ b. This indicates clearly that the selection of a design that is good in all attributes is a complex engineering trade-off. Using the methodology proposed here, one can sort the database of candidate designs to identify a design that is acceptable on all four attributes P mesh, TE, σ c, and σ b while it might not necessarily be the best in any single attribute. Design D marked in Figure 5.3 (and listed in Table 5.2) represents such a design. This design was obtained by searching through the 13,500 candidate designs such that P mesh < 0.3 kw, σ b < 235 MPa, σ c < 1.175 GPa, and the peak-to-peak TE < 0.25 µm. 107

All the candidate designs used in Figure 5.1 to 5.3 used a profile crown of M P = 15 µm and a lead crown of M L = 12 µm. Input torque and speed were kept at T in = 250 Nm and Ω in = 4000 rpm. Design D from Figure 5.3 and Table 5.2 was used next to determine the combined influence of tooth modifications and input torque in the form of M P, M L, and T in on the efficiency, noise and durability metrics. Micro-geometry studies were performed by varying M P and M L between 0 and 30 µm for various T in values. Figure 5.4 shows variation of P mesh with M P and M L at three of these discrete T in values. Design D from Figure 5.3 is also marked in this figure. Here, regardless of T in, increases in M P help reduce P mesh with severe consequences on TE as shown in Figure 5.5. Likewise, the corresponding σ c and σ b in Figure 5.6 and 5.7 are shown to increase with M P and M L further suggesting that the efficiency concerns must be balanced by the other attributes in defining micro-geometry. 108

Candidate Designs Parameter A B C D N 49 28 49 32 Φ [deg] 30 30 22.5 30 Ψ [deg] 27 19 30 19 Λ wd 1.85 1.85 2.504 2.068 d o [mm] 94.16 96.44 95.15 96.42 Root diameter [mm] 88.01 85.01 87.05 85.23 m [mm] 1.6638 3.0898 1.6172 2.7036 Face width [mm] 25 25 25 25 Table 5.2 Gear parameters of the four different designs identified in Figure 5.3. 109

1 0.8 [ ] P mesh kw 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 TE [ µ m] Figure 5.3 Variation of P mesh with (a) peak-to-peak TE, (b) maximum σ c, and (c) maximum σ b. (Continued) 110

Figure 5.3 Continued. 1 0.8 [ ] P mesh kw 0.6 0.4 0.2 0 1000 1100 1200 1300 1400 1 σ c [ MPa] 0.8 P mesh [ kw ] 0.6 0.4 0.2 0 100 200 300 400 500 600 σ b [ MPa] 111