Design and Analysis of a Spiral Bevel Gear

by

Matthew D. Brown

An Engineering Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER OF ENGINEERING IN MECHANICAL ENGINEERING

Approved:

_________________________________________

Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute

Hartford, Connecticut

August, 2009

ii

CONTENTS

Design and Analysis of a Spiral Bevel Gear ...................................................................... i

LIST OF TABLES ............................................................................................................. v

LIST OF FIGURES .......................................................................................................... vi

LIST OF SYMBOLS ....................................................................................................... vii

ACKNOWLEDGMENT .................................................................................................. xi

ABSTRACT .................................................................................................................... xii

1. Introduction .................................................................................................................. 1

2. Gear Theory and Design Methodology ....................................................................... 6

2.1 Material Selection .............................................................................................. 6

2.2 The Material Processing of a Gear ..................................................................... 8

2.2.1 Heat Treatment ....................................................................................... 8

2.2.2 Surface Hardening Treatment (Case Hardening) ................................. 12

2.2.3 Tempering ............................................................................................ 13

2.3 Design of Gear Teeth ....................................................................................... 14

2.4 Loading ............................................................................................................ 17

2.5 Analytical Methodology................................................................................... 19

2.6 Gear Life Calculations ..................................................................................... 22

2.7 Selection of Design Factors ............................................................................. 23

3. Results and Discussion .............................................................................................. 28

3.1 Fatigue Analysis ............................................................................................... 30

3.2 Static Analysis .................................................................................................. 38

3.3 Calculation of Hertz Stresses (Pitting Resistance) ........................................... 38

3.4 Calculation of Bending Stresses ...................................................................... 41

3.5 Gear Life Calculations ..................................................................................... 45

4. Conclusion ................................................................................................................. 46

5. References .................................................................................................................. 48

iii

6. Appendix A ................................................................................................................ 49

7. Appendix B ................................................................................................................ 50

8. Appendix C ................................................................................................................ 51

9. Appendix D ................................................................................................................ 52

iv

Proprietary Information

Warning:

THIS DOCUMENT, OR AN EMBODIMENT OF IT IN ANY MEDIA, DISCLOSES

INFORMATION WHICH IS PROPRIETARY, IS THE PROPERTY OF SIKORSKY

AIRCRAFT CORPORATION, IS AN UNPUBLISHED WORK PROTECTED UNDER

APPLICABLE COPYRIGHT LAWS, AND IS DELIVERED ON THE EXPRESS

CONDITION THAT IT IS NOT TO BE USED, DISCLOSED, OR REPRODUCED, IN

WHOLE OR IN PART (INCLUDING REPRODUCTION AS A DERIVATIVE

WORK), OR USED FOR MANUFACTURE FOR ANYONE OTHER THAN

SIKORSKY AIRCRAFT CORPORATION WITHOUT ITS WRITTEN CONSENT,

AND THAT NO RIGHT GRANTED TO DISCLOSE OR SO USE ANY

INFORMATION CONTAINED THEREIN. ALL RIGHTS RESERVED. ANY ACT

IN VIOLATION OF APPLICABLE LAW MAY RESULT IN CIVIL AND CRIMINAL

PENALTIES.

v

LIST OF TABLES

Table 1 - Typical heat treatments and associated steel grades [5] ..................................... 6

Table 2 - Common SAE steel designations and their nominal alloy contents [5] ............. 7

Table 3 - Overload factors [8] ......................................................................................... 23

Table 4 - Calculated gear tooth loads and bearing reaction loads ................................... 29

Table 5 - Calculated values at critical section A-A ......................................................... 32

Table 6 - Design properties of locking nut ...................................................................... 36

Table 7 - Calculated values at critical section B-B .......................................................... 37

Table 8 - Results for calculating the load sharing ratio and geometry factor .................. 41

Table 9 - Assumed values for θ and its effect on bending stress ..................................... 43

Table 10 - Allowable stress values [8] ............................................................................ 44

vi

LIST OF FIGURES

Figure 1 - Spiral bevel gear mesh [3] ................................................................................ 1

Figure 2 - Input assembly of intermediate gearbox ........................................................... 3

Figure 3 - Case hardenability of carburizing grades of steel [5] ....................................... 7

Figure 4 - Phase diagram of carbon steel [1] ..................................................................... 9

Figure 5 - Electron micrographs of (a) pearlite, (b) bainite and (c) martensite (x7500) [1]

......................................................................................................................................... 10

Figure 6 - Hardenability curves for several steels [1] ...................................................... 11

Figure 7 - Bevel gear nomenclature in the axial plane [3] .............................................. 16

Figure 8 - Bevel gear nomenclature, mean section A-A in Figure 7 [3] ......................... 17

Figure 9 - Loads acting on gear ....................................................................................... 19

Figure 10 - Detailed location of loading .......................................................................... 29

Figure 11 - Location of critical sections .......................................................................... 30

Figure 12 - Constant-life fatigue diagram for heat-treated AISI 4340 alloy steel, Ftu =

150 ksi, Kt = 1.0 [10] ....................................................................................................... 32

Figure 13 - Volume of stressed material for shaft subjected to rotating bending [10] .... 33

Figure 14 - Size effect factor as a function of the volume ratio [10] ............................... 34

Figure 15 - Iterative procedure to calculate the load sharing ratio, mN [10] .................... 40

Figure 16 - Iterative procedure to calculate tooth form factor, Xn [10] ........................... 42

vii

LIST OF SYMBOLS

Symbol Terms Units

A Mean cone distance in

Ao Outer cone distance in

Ar Area in2

a Mean addendum in

ao Larger end addendum in

aog Gear addendum in

aop Pinion addendum in

at Thread pitch diameter in

bo Larger end dedendum in

bog Gear dedendum in

bop Pinion dedendum in

C Clearance in

CP Circular pitch in

Ci Inertia factor -

Cm Load distribution factor -

Co Overload factor -

Cp Elastic coefficient (lbs/in2).5

Cv Dynamic factor -

c Mean collar diameter of nut in

Di Inner diameter in

Do Outer diameter in

d Pitch diameter in

dg Gear pitch diameter in

dp Pinion pitch diameter in

dog Gear outside diameter in

dop Pinion outside diameter in

E Young's modulus lb/in2

F Face width in

F' Net face width in

Fe Effective face width in

Fen Adjusted endurance limit lb/in2

Fen' Endurance limit at 10^8 cycles corrected for steady stress lb/in2

Fk Projected length of s contained within the tooth bearing ellipse in the lengthwise direction in

Fr Reliability factor -

viii

Fs Size effect factor -

Ftu Ultimate tensile strength lb/in2

f Distance from the midpoint of the tooth to the line of action in

fa Normal stress lb/in2

fb Bending stress lb/in2

fc Compressive stress lb/in2

fs Steady torsion lb/in2

fsteady Principle steady stress lb/in2

fv Vibratory stress lb/in2

fvib Vibratory bending lb/in2

HP Horsepower hp

hk Working depth in

ht Whole depth in

I Geometry factor for compressive stress -

I.D. Maximum inner diameter in

J Geometry factor for bending stress -

K Torque coefficient of nut -

Kf Actual stress concentration factor -

Kfs Surface finish factor -

Ki Inertia factor for I -

Km Load distribution factor -

Ks Size factor -

Kt Theoretical stress concentration factor -

K* Correlation factor -

k Total number of different stress levels -

l Lead in/thd

M Bending moment lb in

M.S. Margin of safety -

mf Face contact ratio -

mn Load sharing ratio -

mo Modified contact ratio -

mp Transverse (profile) contact ratio -

N Number of threads per inch thd/in

Nfi Total number of cycles to failure at i-th stress level -

Ng Number of teeth in gear -

Np Number of teeth in pinion -

ni Number of cycles at i-th stress level -

ix

O.D. Minimum outer diameter in

P Axial Load lb

PITCH Diametral pitch in-1

PN Mean normal base pitch in

Pn Mean normal circular pitch in

p Large end transverse circular pitch in

pn Mean normal circular pitch in

p3 Distance in mean normal section from beginning of action to point of load application in

R Mean transverse pitch radius in

RPM Revolutions per minute rpm

Rbng Mean normal base radius of gear in

Rbnp Mean normal base radius of pinion in

Rg Mean transverse pitch radius of gear in

Rng Mean normal pitch radius of gear in

Rnp Mean normal pitch radius of pinion in

Rong Mean normal outside radius of gear in

Ronp Mean normal outside radius of pinion in

Rp Mean transverse pitch radius of pinion in

Rt Mean transverse radius to point of load application in

Rx Radius in mean normal section to point of load application on the tooth centerline in

rf Fillet radius in

rt Cutter edge radius in

s Length of line of contact in

T Torque lb in

Tg Gear torque lb in

Tp Pinion torque lb in

t Stress in numbers of standard deviations from the mean -

tn One half the tooth thickness at the critical section of the gear tooth in

to Large end circular tooth thickness in

*tog Pinion circular thickness in

V Volume of critically stressed material in3

V.R. Volume ratio -

Vcr Volume ratio of critically stressed material in3

Wa Axial thrust lb

Wr Seperating load lb

Wt Tangential tooth load lb

Wtg Gear tangential tooth load lb

x

Wtp Pinion tangential tooth load lb

Xn Gear tooth strength ratio -

Xo Gear pitch apex to crown in

Xo" Distance from mean section measured in the lengthwise direction along the tooth in

xo Pinion pitch apex to crown in

Yk Tooth form factor -

Z Section modulus in3

Zn Length of action in mean normal section in

αg Gear addendum angle deg

αp Pinion addendum angle deg

Γ Gear pitch angle deg

ΓR Gear root angle deg

Γo Gear face angle of blank deg

γ Pinion pitch angle deg

γo Pinion face angle of blank deg

γR Pinion root angle deg

δg Gear dedendum angle deg

δp Pinion dedendum angle deg

ΔFH' Heel increment -

ΔFT' Toe increment -

θ Pressure flank angle deg

μ Poisson's ratio -

μf Coefficient of friction -

ν Coefficient of variation -

π Pi -

ρ Profile radius of curvature at pitch circular in mean normal section in

ρr Minimum fillet radius in

ρo Relative radius of curvature in

Σ Shaft angle deg

Ф Pressure angle deg

Фh Pressure angle at point of load application deg

Фn Angle at which the normal force makes with a line perpendicular to the tooth centerline deg

ψ Mean spiral angle deg

Ψb Base spiral angle deg

xi

ACKNOWLEDGMENT

First and foremost, the author wishes to thank his family and friends who have supported

him throughout his life. He also wishes to thank all of his previous and current academic

inspirations that have guided him to this point in his academic career. Lastly, the author

would like to thank his peers in the transmission department at Sikorsky Aircraft who

were always willing to share their extensive knowledge of gear design and analysis.

xii

ABSTRACT

This investigation gives a detailed approach to spiral bevel gear design and

analysis. Key design parameters are investigated in accord with industry standards and

recommended practices for use in a medium class helicopter. Potential gear materials

are described leading to the selection of SAE 9310 steel as the proper material for this

application, finished with carburization and case hardening processes. A final gear

design is proposed and analyzed to show that proper margins of safety have been

included in the design. Fatigue analysis is conducted at the two most critical sections of

the gear shaft resulting in margins of safety equal to .48 and 3.35. Static analysis is

conducted at the most critical section in accordance with Federal Aviation

Administration requirements, resulting in a margin of safety equal to .87. Further

analysis is conducted on the gear teeth to ensure proper gear tooth geometry and proper

loading techniques. Hertz stresses are investigated and calculated to be 180.6 ksi which

allows for proper resistance to pitting. Bending stresses are calculated equal to 31.5 ksi

which shows proper bending strength in the gear teeth to mitigate the risk of failure to a

gear tooth. Results are compared to the recommended allowable stresses as published

by the American Gear Manufacturing Association. Finally, fatigue life calculations are

performed to show that the gear has been designed with unlimited life for this specific

application.

1

1. Introduction

A gear is a mechanical device often used in transmission systems that allows

rotational force to be transferred to another gear or device. The gear teeth, or cogs,

allow force to be fully transmitted without slippage and depending on their

configuration, can transmit forces at different speeds, torques, and even in a different

direction. Throughout the mechanical industry, many types of gears exist with each type

of gear possessing specific benefits for its intended applications. Bevel gears are widely

used because of their suitability towards transferring power between nonparallel shafts at

almost any angle or speed. Spiral bevel gears have curved and sloped gear teeth in

relation to the surface of the pitch cone. As a result, an oblique surface is formed during

gear mesh which allows contact to begin at one end of the tooth (toe) and smoothly

progress to the other end of the tooth (heel), as shown below in Figure 1. Spiral bevel

gears, in comparison to straight or zerol bevel gears, have additional overlapping tooth

action which creates a smoother gear mesh. This smooth transmission of power along

the gear teeth helps to reduce noise and vibration that increases exponentially at higher

speeds. Therefore, the ability of a spiral bevel gear to change the direction of the

mechanical load, coupled with their ability to aid in noise and vibration reduction, make

them a prime candidate for use in the helicopter industry.

Figure 1 - Spiral bevel gear mesh [3]

Toe

Heel

Toe

Heel

2

The American Gear Manufacturing Association (AGMA) has developed

standards for the design, analysis, and manufacture of bevel gears. The first step in any

general design employing gears is to first predict and understand all of the conditions

under which the gears will operate. Most importantly are the anticipated loads and

speeds which will affect the design of the gear. Additional concerns are the operating

environment, lubrication, anticipated life of operation, and assembly processes, just to

name a few.

The spiral bevel gear designed and analyzed herein will be utilized on an upgrade

program for an existing helicopter firmly established in the medium class commercial

helicopter industry. It will be modeled after a gear that has been operating in the

intermediate gearbox of this helicopter for over 25 years and has logged over 5 million

flight hours. The new gear will have a reduced number of teeth, from 28 teeth to 26

teeth, in order to reduce the speed of the tail rotor. The slower tail rotor speed will allow

the tail rotor blades to operate more efficiently, in the hopes of reducing both vibration

and noise caused by the tail rotor. Ultimately, this will result in a quieter helicopter that

operates more smoothly than previous models. It is important to note that further design

improvements of the intermediate gearbox in which this gear will operate are not being

implemented. The bearings, liners, seals, and transmission housings are not changing

and therefore the general design envelope for the gear has not changed either.

The intermediate gearbox transmits torque between two drive shafts at an angle

of 57 degrees, and reduces the speed from 3491 RPM to 3130 RPM while operating at

full speed. The speed reduction is a result of the gear mesh between two spiral bevel

gears, the pinion possessing 26 teeth and the gear possessing 29 teeth, thereby producing

a reduction ratio of 1 to .905. The intermediate gearbox weighs about 22.2 pounds when

filled with approximately .260 gallons of oil, which lubricates the gears by splash

lubrication – an oil pump or jet powered lubrication system is not necessary. When

filled with oil, the gear mesh occurs above the oil line, but as the gears rotate out of

mesh, they dip through the oil and lubricate the gear teeth prior to meshing. This oil film

prevents scuffing and scoring of the gear teeth and helps reduce the friction and heat

generation caused by the clash of the gear teeth. Centrifugal forces, an outward force

3

associated with rotation, also play a big part in splash lubricating the gear mesh. As the

gears rotate through the oil, the centrifugal force flings the oil against the walls of the

center transmission housing. This housing was designed with a reservoir at the top of

the housing, in order to capture any of the oil flung during rotation. Oil that collects in

this reservoir then drips through a drain hole directly onto the gear mesh or through two

drip ports to lubricate the outer bearing of both the input and output assemblies. The

configuration of the input assembly of the intermediate gearbox can be seen in Figure 2

below.

Figure 2 - Input assembly of intermediate gearbox

The way in which a gear will be loaded is given the utmost attention during the

design process. Based on AGMA recommendations, the following load conditions are

considered: the power rating of the prime mover, its overload potential and the

uniformity of its output torque; the output loading including the normal output load,

peak loads and their duration; the possibility of stalling or severe loading at infrequent

intervals; and inertia loads arising from acceleration or deceleration [3]. An

understanding of these load conditions allows for basic load calculations and the

4

selection of suitable safety factors in order to obtain protection for expected intermittent

overloads, desired life expectancy, and safety.

In applications where gears will experience peak loads, such as during normal

helicopter operation, the most important consideration is given to the allowable duration

of peak loads. The AGMA recommends that if the total duration exceeds ten million

cycles during the total expected life of the gear, the value of the peak load is to be used

for estimation of the size of the gear. For peak loads whose duration is less than ten

million cycles, a value equal to one half the value of the peak load or the highest

sustained load, whichever is greater, is to be used for estimation of gear size [3]. These

recommendations are based on not knowing the complete flight spectrum and therefore

introduce a conservative approach into the design effort. Fortunately, in this application,

the full flight spectrum is known because of the long service history of this helicopter. A

wide ranging flight spectrum has been established that displays the changing speeds,

horsepower, and torque values depending on the maneuver of the helicopter at any point

in time. Appendix A shows the applicable flight spectrum in which this gear will

operate. Two data sets are presented in this appendix, recorded flight data from the

flight test program in which the previous gear operated, and the estimated flight

spectrum that is anticipated for the upgraded model of the helicopter. The estimated

flight spectrum is based on analytical tools proprietary to Sikorsky Aircraft and therefore

will not be discussed here in detail. It does however estimate performance parameters of

the helicopter and the loads that will occur in flight based on the overall design of the

helicopter. The estimated loads for this application are shown in Appendix A in the

column titled “New Design”. These loads are presented as torque values, so in order to

get an accurate understanding of the loads transmitted by the gear, the torque values are

converted to horsepower. This can be done using the formula [7],

𝐻𝑃 =𝑇 ∗ 𝑅𝑃𝑀

5252

Equation 1

5

where T is the applied torque and RPM is the operating speed of the gear shaft. The

converted horsepower values are shown in column P of Appendix A. A brief review of

the data shows that a maximum peak load of 346 horsepower is expected during Regime

#58, which is a transient condition, or peak load, that occurs 20 times per 100 hundred

flight hours, or .03% of the life of the aircraft. The normal operating condition however,

Regime #10, which occurs 30% of the time and induces only 34 horsepower, is much

lower than the peak load. A detailed review of the data shows that the estimated loads

for the new gear application are much lower than the previous application such that if the

previous load spectrum is used, the design will be that much more conservative. Also,

limiting the design changes to only the necessities, allows the other components of the

input assembly, shown previously in Figure 2, to still be used which is the ultimate goal

of this redesign effort. As a result, because the previous gear application was designed

for higher loads which included a normal operating condition of 240 horsepower, this

value will also be used throughout the analysis to incorporate additional conservatism

into the design of the gear.

6

2. Gear Theory and Design Methodology

2.1 Material Selection

The specific application of a gear determines the necessary material properties and

additional treatments that may be required. Additional treatments typically considered

are through hardening and surface hardening, which includes but is not limited to

carburization, nitriding, induction hardening, and flame hardening. Through hardened

steels are used when medium wear resistance and load carrying capacity are desired

whereas carburized and hardened gears are used when high wear resistance and high

load carrying capacity are required [5]. Specifically, the desired loading and desired

design life are integral in selecting the proper material and any additional treatment that

may be required.

Many years of gear industry experience has led the design community to rely on

carburized, case-hardened steel for bevel gears. Testing has been performed on these

types of materials and allowable stresses have been derived as a result of these widely

recognized test results. Therefore, spiral bevel gear materials are limited to only those

which are easily carburized and case-hardened. Table 1 below, generated from AGMA

recommendations of associated steel grades and their typical heat treatments, displays

the seven potential steel grades which are recognized to be well suited towards

carburization in bevel gear applications.

Table 1 - Typical heat treatments and associated steel grades [5]

Heat Treatment Steel Grade

Carburizing

1020

4118

4320

4820

8620

8822

9310

18CrNiMo7-6

7

To better understand the steel grades above and their metallurgical compositions, Table

2 below shows the common steel designations and their nominal alloy contents.

Table 2 - Common SAE steel designations and their nominal alloy contents [5]

Carbon Steels

10xx No intentional alloying

15xx Mn 1.00 - 1.35%

Alloy Steels

41xx Cr 1%, Mo 0.25%

43xx Ni 1.75%, Cr 0.75%, Mo 0.25%

86xx Ni 0.5%, Cr 0.5%, Mo 0.2%

93xx Ni 3.25%, Cr 1.25%, Mo 0.12%

Note: "xx" = (nominal percent carbon

content x 100)

Steels under consideration also must have sufficient case hardenability in order to obtain

adequate hardness below the depth of the carburized case. Figure 3 below shows the

case hardenability for the alloy steels shown in Table 1 and Table 2.

Figure 3 - Case hardenability of carburizing grades of steel [5]

8

The horizontal axis of Figure 3 is the ruling section, also called the controlling section,

measured in millimeters (mm). The controlling section is defined as the section size of

the gear which has the greatest effect in determining the rate of cooling during

quenching, measured at the location where the specified hardness is required [5]. The

controlling section for the gear discussed throughout this paper has a diametral

measurement equal to approximately seven inches, or two hundred millimeters. Using

this value eliminates SAE 41xx, 86xx, and 88xx series steels from consideration because

they will not case harden adequately at the diameter measured for the ruling section.

SAE 9310 will provide the most adequate case hardenability, according to Figure 3, and

therefore will be selected as the material from which this gear will be manufactured.

2.2 The Material Processing of a Gear

2.2.1 Heat Treatment

Carbon steel exists in a mechanical mixture of two primary metallurgical phases,

a dilute alloy of the element iron in a form metallurgically known as ferrite, and the

chemical compound iron carbide in a form metallurgically known as cementite. An

important third microconstituent is a microcomposite consisting of cementite platelets

embedded in ferrite, which is called pearlite as a result of its mother-of-pearl appearance

under magnification. On occasion, a secondary metallurgical form will also be present,

called bainite, another mixture of carbide and ferrite. When carbon steel of this nature is

heated above its lower critical point, frequently in the range of 1,100 to 1,200 degrees

Celsius, the layers of ferrite and cementite that make up the pearlite begin to merge into

each other until the pearlite is thoroughly dissolved, forming what is known as austenite

[6]. If the steel reaches its upper critical point, the combination of ferrite and cementite

will be fully converted to austenite. This can be seen in the phase diagram of carbon

steel, shown below in Figure 4.

9

Figure 4 - Phase diagram of carbon steel [1]

Once the full transformation of pearlite to austenite has been accomplished, the

carbon steel can be cooled to form various crystalline structures which will greatly alter

the material properties of the steel. A slow rate of cooling will transform the austenite

back to pearlite whereas a rapid rate of cooling, termed the critical cooling rate, will

cause the austenitized steel to form a new structure called martensite. This

microstructure is characterized by an angular needlelike structure and a very high

hardness, thereby making it highly desirable in applications where high wear resistance

and load carrying capacity are required. Figure 5 below illustrates the difference in size

and shape of the microstructures of pearlite, bainite, and martensite at a magnification of

x7500.

10

Figure 5 - Electron micrographs of (a) pearlite, (b) bainite and (c) martensite

(x7500) [1]

More specifically, the relationship between a steels mechanical properties and the

cooling rate that governs them is a qualitative measure of hardenability. Since hardness

is directly related to the amount of martensite in the sample, hardenability measures the

ability of the steel to harden as a result of quenching. A typical hardenability curve is

useful in determining how much martensite, for a given rate of cooling, will replace

pearlite and bainite during the cooling process. Figure 6 below displays a hardenability

curve for several alloy steels.

11

Figure 6 - Hardenability curves for several steels [1]

Using an industry standard, the Jominy distance test, Figure 6 shows that hardness

decreases with increasing distance from the quench surface. Therefore, steel regarded as

highly hardenable will retain large values of hardness for relatively long distances. The

important feature of the figure above is the varying curve shapes displayed by the alloy

steels. The 1050 and 4320 alloys show a shallow depth of hardness below the surface

whereas the other four alloys exhibit a high hardness persisting to a much greater depth.

On the other hand, the selected material of SAE 9310 is almost that of a plateau,

showing that it will retain a high hardness value throughout the specimen. The disparity

in curve shapes can best be attributed to the content of nickel, chromium, and

molybdenum in the specific alloys. These alloying elements delay the austenite-to-

pearlite and/or bainite reactions which permits more martensite to form for a particular

cooling rate, yielding a greater hardness [2]. The maximum attainable hardness of any

steel is only realized when the cooling rate in quenching is rapid enough to ensure full

transformation to martensite [6].

12

In addition to the hardenability characteristics discussed above, the cooling rate

of a specimen can greatly affect the resulting hardness. Because the cooling rate

depends on the rate of heat extraction from the specimen, factors such as size, geometry,

type and velocity of quenching medium all have an immediate effect on the resulting

hardness. Of the three most popular quenching mediums, water, oil and air, oil is the

most suitable for heat treatment of most alloy steels as water is often too severe and

results in cracking or warping of a specimen. Air quenching often results in a pearlitic

structure and is therefore ineffective in obtaining the desired martensitic structure.

Geometry also affects the rate at which heat energy is dissipated to the quenching

medium. The relationship to cooling rate is often determined by ratio of surface area to

the mass of the specimen. The larger this ratio, the more rapid will be the cooling rate

and, consequently, the deeper the hardening effect [2].

In summary, the ultimate goal of a heat treatment procedure is to convert weaker

metallurgical grain structures such as pearlite and bainite to a stronger structure like

martensite. This process is typically performed using two essential steps; heating the

steel to some temperature above its transformation point such that it becomes entirely

austenitic in structure, and then quenching the steel at some rate faster than the critical

rate in order to produce a martensitic structure. The resulting martensitic structure is

mainly dependent on three factors (1) the composition of the alloy (austenite grain size

and prior microstructure), (2) the type and character of the quenching medium (time and

temperature during austenitizing), and (3) the size and shape of the specimen [2]. Water,

oil and air can be used to increase the rate of cooling, but oil is by far the most effective

quenching medium when attempting to form a fully martensitic structure. Geometry and

shape of a specimen can also affect the resulting microstructure after quenching, and

therefore it is important to investigate the rate at which hardness drops off with distance

into the interior of a specimen as a result of diminished martensite content [2].

2.2.2 Surface Hardening Treatment (Case Hardening)

Low carbon steel, typically containing .10 to .20 percent of carbon, can be further

hardened at its surface by impregnating a component’s outer surface with a sufficient

13

amount of carbon. This process, termed carburization, is a solution to applications

which require high hardness or strength primarily at the surface, but also core strength

and toughness to withstand impact stress. The carburized parts are later heat treated in

order to obtain a hard outer case and, at the same time, give the core the required

physical properties. The term “case hardening” is ordinarily used to indicate the

complete process of carburizing and hardening [6]. This paper will focus on carburizing

because that treatment has been chosen to be best suited for this gear application.

During the carburizing process, carbon is diffused into the part’s surface to a

controlled depth by heating the part in a carbonaceous medium. The most commonly

used mediums include liquid carburizing, which involves heating the steel in molten

barium cyanide or sodium cyanide; gas carburizing, which involves heating the steel in a

gas of controlled carbon content; and pack carburizing, which involves sealing both the

steel and solid carbonaceous material in a gas-tight container, then heating this

combination [6]. The case depth, or resulting depth of carburization, is dependent upon

the carbon potential of the medium used and the time and temperature of the

carburization treatment. Temperatures typically range from 1,550 to 1,750 degrees

Farenheit with the temperature adjusted to obtain specific case depths for the intended

application. Carburizing the entire part is typically not necessary, as it is only required

where high hardness at the surface is necessary, for example gear teeth, spline teeth, and

bearing journals. Sections of the gear that are not to be carburized are usually covered

with copper plating which prevents the carbon from diffusing into the surface of the

specimen in areas where the copper plate is applied. A standard case hardening

procedure allows for the carburizing cycle to occur prior to quenching, thereby reducing

the need for reheating.

2.2.3 Tempering

Steel that has been converted to a martensitic structure by sudden cooling in a

quenching bath, such as a heat treated steel, often becomes brittle and forms undesired

internal strains. The brittleness and internal strains must be removed prior to machining

so as to avoid fast fracture to the work piece. In order to remove the brittleness and

14

internal strains, the work piece is heated to about 300 to 750 degrees Farenheit, which

softens the gear and releases the preexisting strains. This process is called tempering

and may include heating to even 750 to 1290 degrees Farenheit depending on how

ductile the work piece needs to be before proper machining can take place. It may also

be used to alter toughness, to refine crystal structure and grain orientation, or to relieve

stress or hardness from a working surface, all of which allow the gear to be more easily

machined during later operations.

2.3 Design of Gear Teeth

The process of designing gear teeth is somewhat arbitrary in that the specific

application in which the gear will be used determines many of the key design

parameters. Recommended design practices are published in the AGMA standard 2005-

D03, Design Manual for Bevel Gear Teeth. This design standard illustrates all aspects

of bevel gear tooth design, starting from preliminary design values and progressing

towards a finished design ready to be analyzed. Not only does it give recommended

practices for design, it also covers manufacturing considerations, inspection methods,

lubrication, mounting methods, and appropriate drawing formats. While this is certainly

an invaluable tool published in order to provide one guideline for the design of bevel

gears across all industries, it does not always properly differentiate design parameters

that should be used for one industry versus another. For example, the automotive

industry typically uses cast iron gears in transmission applications whereas a cast iron

gear would not be feasible in a helicopter transmission because of the high level of

loading and occurrence of peak loads that have the potential to be significantly higher

than the load at normal operating conditions. As a result, a gear designer must have

significant experience in the appropriate industry and be able to make intelligent

decisions based on the specific application, which may or may not agree with the

AGMA recommendations. In addition, many of the recommendations are based on

spiral bevel gears meshing at a shaft angle of 90 degrees, whereas in this application, the

bevel gears mesh at a shaft angle of 57 degrees.

15

Spiral angle and pressure angle are two design parameters that help determine the

shape of a spiral bevel gear tooth. Common design practices have determined that for

spiral bevel gears, a pressure angle of twenty degrees and a spiral angle of thirty five

degrees should be used. Following this common practice for selection of spiral angle

establishes a good face contact ratio which maximizes smoothness and quietness during

gear mesh. In regards to the selection of a pressure angle, a lower pressure angle

increases the transverse contact ratio, a benefit which results in increased bending

strength, while also increasing the risk of undercut which is a major concern. Lower

pressure angles also help to reduce the axial and separating forces and increase the

toplands and slot widths. These factors help to strengthen the gear teeth because the

increased slot widths allow the use of larger fillet radii, resulting in increased bending

strength. The contact stress is reduced however, as a result of the larger fillet radii, so

close consideration is required to ensure the correct pressure angle is chosen for the

intended application.

In addition, spiral bevel gears are designed such that the axial thrust load tends to

move the pinion out of mesh. This helps to avoid the loss of backlash, defined as the

clearance between mating components. While a lot of backlash is not desirable, small

amounts of backlash are required to allow for proper lubrication, manufacturing errors,

deflection under load, and differential expansion between the gears and housing.

As previously mentioned, the gear addressed throughout this paper is replacing a

similar gear that operated in the fleet for many years. The main difference between the

two gears is the number of teeth on the pinion which helps to achieve the proper gear

reduction ratio to reduce the speed at the tail rotor. As a result, the design of this gear

was simplified because not everything had to be developed from scratch. Important

geometric design parameters remained constant between the old gear and the new gear in

order to be able to use the existing bearings, transmission housings, seals, and other

hardware. Minimal changes were made to the values for diametral pitch, pitch diameter,

pitch angles, and face width, which are the basis for calculating the necessary geometric

16

design parameters shown in Appendix C. These parameters are shown in Figure 7 and

Figure 8 below.

Figure 7 - Bevel gear nomenclature in the axial plane [3]

17

Figure 8 - Bevel gear nomenclature, mean section A-A in Figure 7 [3]

2.4 Loading

Torque application to a spiral bevel gear mesh induces tangential, radial, and

separating loads on the gear teeth. For simplicity, these loads are assumed to act as point

loads applied at the mid-point of the face width of the gear tooth. The radial and

separating loads are dependent upon the direction of rotation and hand of spiral, in

addition to pressure angle, spiral angle and pitch angle. The tangential loads are defined

as [10],

𝑊𝑡𝑝 =2𝑇𝑝

𝑑𝑝 − 𝐹 sin 𝛾

Equation 2

for the pinion and,

18

𝑊𝑡𝑔 =2𝑇𝑔

𝑑𝑝 − 𝐹 sin Γ

Equation 3

for the gear, with T equal to the torque, dp equal to the pitch diameter, F equal to the face

width, γ equal to the pitch angle of the pinion , and Γ equal to the pitch angle of the gear.

The radial and separating loads are calculated as a percentage of the tangential loads

calculated above. For a right hand of spiral rotating counter clockwise, the axial thrust

load for a driving member (pinion) is defined as [10],

𝑊𝑎 =𝑊𝑡

cos 𝜓 tan 𝜙 sin 𝛾 + sin 𝜓 cos 𝛾

Equation 4

and the separating load is defined as,

𝑊𝑟 =𝑊𝑡

cos 𝜓 tan 𝜙 cos 𝛾 − sin 𝜓 sin 𝛾

Equation 5

where Φ is the pressure angle, and ψ is the mean spiral angle. Figure 9 below displays

the line of action through which the tangential, axial and separating loads act.

19

Figure 9 - Loads acting on gear

The loads in Figure 9 above, labeled RBA, RBH, RBV, RAA, RAV, and RAH, are reaction

loads generated by the two tapered roller bearings that support the gear shaft. It is the

responsibility of these bearing reaction loads to counteract the forces generated by the

mesh of the gear teeth, shown in the figure as Wap, Wtp and Wrp. A detailed view of the

gear assembly was previously shown in Figure 2.

2.5 Analytical Methodology

Spiral bevel gear teeth are primarily designed for resistance to pitting and for

their bending strength capacity. This is not to say that other types of gear tooth

deterioration such as scuffing, wear, scoring, and case crushing are of less importance,

but proper design techniques established to employ designs for pitting resistance and

bending strength will often result in gears that are not affected by additional types of

tooth deterioration. Design for pitting resistance is primarily governed by a failure mode

20

of fatigue on the surface of the gear teeth under the influence of the contact stress

between the mating gears [7]. Design for bending strength capacity is based on a failure

mode of breakage in the gear teeth caused by bending fatigue.

Pitting resistance is related to Hertzian contact (compressive) stresses between

the two mating surfaces of gear teeth. The formulas were developed based on Hertzian

theory of the contact pressure between two curved surfaces and load sharing between

adjacent gear teeth as well as load concentration that may result from uncertainties in the

manufacturing process. The contact stress is mainly a function of the square root of the

applied tooth load. Three primary types of pitting are widely recognized throughout the

industry; they are initial pitting, micropitting, and progressive pitting. Initial pitting

often occurs early in the life of the gear and is not deemed a serious cause for concern.

It is a result of localized overstressed areas and is characterized by small pits which do

not extend over the entire face width or profile depth of the affected tooth [8].

Typically, initial pitting redistributes the applied load by progressively removing high

contact spots, and once these high contact spots are removed, the pitting stops.

Micropitting, also called frosting, is typical in case hardened steels. Unlike

initial pitting, micropitting appears as very small micro-pits, unseen by the naked eye,

and has the potential to cover the entire gear tooth. It appears as a light gray matte finish

on the tooth surface and can most often be attributed to improper surface finish or

lubrication. The third and final type of pitting is progressive pitting, which creates large

surface pits to start and progresses until a considerable portion of the tooth surface has

developed pitting craters of various shapes and sizes. The dedendum section of the drive

gear (pinion), shown previously in Figure 8, is often the first to experience serious

pitting damage and could potentially be the point of initiation of a bending fatigue crack,

causing a tooth breakage failure [8].

The basic equation for compressive stress in a bevel gear tooth is given by [10],

𝑓𝑐 = 𝐶𝑝 𝑊𝑡 × 𝐶𝑜

𝐶𝑣

1

𝐹 × 𝑑𝑝

𝐾𝑚

𝐼

Equation 6

21

where Cp is the elastic coefficient, Wt is the tangential tooth load, Co is the overload

factor, Cv is the dynamic factor, F is the face width, dp is the pitch diameter, Km is the

load distribution factor, and I is the geometry factor. The elastic coefficient is defined as

[10],

𝐶𝑝 = 3

4𝜋

𝐸

(1 − 𝜇2)

Equation 7

where E is the Young’s modulus of the material and µ is Poisson’s ratio. For almost all

steels, including SAE 9310 the type used for this gear application, the values for E and µ

are 3 x 107 psi and .30 respectively.

Bending strength capacity ratings in bevel gear teeth are developed using a

simplified approach to cantilever beam theory. This methodology accounts for various

factors including: the compressive stresses at the tooth roots caused by the radial

component of the tooth load; the non-uniform moment distribution of the load resulting

from the inclined contact lines on the teeth of spiral bevel gears; stress concentration at

the tooth root fillet; load sharing between adjacent contacting teeth; and lack of

smoothness due to low contact ratio [8]. Calculating the bending strength rating will

determine the acceptable load rating at which tooth root fillet fracture should not occur

during the entirety of the life of the gear teeth under normal operation.

The basic equation for bending stress in a bevel gear is given by [10],

𝑓𝑏 = 𝑊𝑡 𝑃𝐼𝑇𝐶𝐻

𝐹

𝐾𝑠 × 𝐾𝑚

𝐽

Equation 8

where Wt is the tangential tooth load previously discussed, PITCH is the diametral pitch,

F is the face width, Ks is the size factor, Km is the load distribution factor, and J is the

geometry factor. In this case, the diametral pitch should be taken at the outer end of the

tooth and equal [10],

22

𝑃𝐼𝑇𝐶𝐻 =𝑁𝑝

𝑑𝑝

Equation 9

where Np is the number of teeth on the pinion.

2.6 Gear Life Calculations

Per the recommendations of the AGMA, Miner’s rule is used to calculate the

effects of cumulative fatigue damage under repeated and variable intensity loads.

Miner’s rule is based on the theory that the portion of useful fatigue life used up by a

number of repeated stress cycles at a particular stress is proportional to the total number

of cycles in the overall fatigue life of the part. Using this hypothesis, Miner’s rule

assumes that the damage done by each stress repetition at a given stress level is equal,

and that the first stress cycle at a uniform stress level is as damaging as the last [8].

Because of this, the order in which the individual stress cycles are applied is not

significant.

Based on the methodology of Miner’s rule, loads in excess of the gear’s

endurance limit will cause damage. Failure is to be expected when [8],

𝑛𝑖

𝑁𝑓𝑖

𝑘

𝑖=1

≥ 1.0

Equation 10

where k is the total number of different stress levels, ni is the number of cycles at the i-th

stress level, and Nfi is the total number of cycles to failure at the i-th stress level. A

detailed review of the load spectrum, presented in Appendix A, shows that a total of five

flight maneuvers cause horsepower loads greater than 240HP, the endurance limit of the

gear. Summing the total percent time of each maneuver, it is shown that fatigue damage

to the gear occurs during 1.53% of the estimated flight spectrum. Each of these

horsepower loads are then converted to stress cycles, then to damage accumulation using

Miner’s rule. Both bending life damage and durability life damage are calculated to

23

ensure an adequate fatigue life for bending strength and pitting resistance. Details of this

analysis are presented in Section 3.5.

2.7 Selection of Design Factors

The dynamic factor, Cv, used in calculation of the pitting resistance factor,

accounts for quality of gear teeth while operating at the specified speed and load

conditions. It is typically influenced by design effects, manufacturing effects,

transmission error, dynamic response, and resonance. In a broader sense, the dynamic

factor makes allowance for high-accuracy gearing which requires less derating than low-

accuracy gearing and, at the same time, makes allowance for heavily loaded gearing

which requires less derating than lightly loaded gearing [8]. When gearing is

manufactured using very strict processes and controls, resulting in very accurate gearing,

typical values of Cv between 1.0 and 1.1 are used. For this application, a Cv value of 1.0

will be used.

The overload factor, Co, accounts for momentary peak loads that are much higher

than the normal operating conditions. Typical causes of peak loads in helicopter

applications can be attributed to wind gust loads, system vibration, operation through

critical speeds, overspeed conditions, and braking (application of the rotor brake). Table

3 below, taken from AGMA 2003-B97, provides recommended values for the overload

factor based on characterization of the momentary peak loads that may be experienced.

Table 3 - Overload factors [8]

24

As previously discussed in Section 1, and shown in Appendix A, all peak loads and

normal operating load conditions are known and accounted for. As a result, there is no

need to use an overload factor because the gear has already been designed with peak

loads in mind. Therefore, an overload factor equal to 1.0 will be used.

The load distribution factor, Km, is a function of the rigidity of the mounting and

reflects the degree of misalignment under load. It modifies the rating formulas in order

to capture the non-uniform distribution of the load along the length of the gear tooth.

The amount of non-uniformity of the load distribution is a function of gear tooth

manufacturing accuracy, tooth contact and spacing, alignment of the gear in its

mounting, bearing clearances, and geometric characteristics of the gear teeth, and

therefore all are considerations which affect the load distribution factor [8]. Because the

gear being designed in this application is supported by dual taper roller bearings, the

mounting of the gear is considered rigid which minimizes misalignment between the

gear, the bearings, and their mountings. Misalignment will exist however, as a result of

assembly tolerances, backlash, and manufacturing tolerances. Design experience leads

to a choice of 1.10 for the load distribution factor, Km.

The size factor, Ks, is a reflection of non-uniformity of material properties and is

a function of the strength of the material. In addition to material properties, it depends

primarily on tooth size, diameter of the part, face width, and ratio of tooth size to

diameter of the part. The size factor can be quickly calculated using [10],

𝐾𝑠 =1

𝑃𝐼𝑇𝐶𝐻.25

Equation 11

The geometry factor for resistance to pitting, I, evaluates the effects that the

geometry of the gear tooth has on the stresses applied to the gear tooth. More

specifically, it evaluates the relative radius of curvature of the mating tooth surfaces and

the load sharing between adjacent pairs of teeth at the point on the tooth surfaces where

the calculated contact pressure will reach its maximum value [8]. The geometry factor

may be calculated from [10],

25

𝐼 =𝐴

𝐴𝑜

𝑠

𝐹

𝜌𝑜

𝑑

cos 𝜓

𝐾𝑖

cos 𝜑

𝑚𝑛

Equation 12

where A is the mean cone distance, Ao is the outer cone distance, s is the length of line of

contact, F is the actual face width, ρo is the relative radius of curvature, Ki is the inertia

factor, and mn is the load sharing ratio. Calculation of the geometry factor includes an

iterative process in order to minimize the distance from the mid-point of the tooth to the

line of action because ultimately, it is desired to have the line of action go through the

mid-point of the tooth. Successful minimization of this distance will result in the

smoothest stress distribution across the gear tooth.

The geometry factor for bending strength, J, is also concerned with gear tooth

geometry but gives more consideration to the shape of the tooth and the stress

concentration due to the geometric shape of the root fillet. Careful consideration is also

given to the position at which the most damaging load is applied, the sharing of load

between adjacent pairs of teeth, the tooth thickness balance between the pinion and

mating gear, the effective face width due to lengthwise crowning of the teeth, and the

buttressing effect of an extended face width on one member of the pair [8].

Incorporation of these variables leads to the following definition for the geometry factor

[10],

𝐽 =𝐴

𝐴𝑜

𝑅𝑡

𝑅

𝐹𝑒𝐹

𝑌𝑘

𝑚𝑛 × 𝐾𝑖

Equation 13

where Rt is the mean transverse radius of load application, R is the mean transverse pitch

radius, Fe is the effective face width, and Yk is the tooth form factor.

The inertia factor used for both bending strength and pitting resistance, Ki, can be

determined from [10],

𝐾𝑖 = 𝐶𝑖 =2.0

𝑚𝑜 𝑖𝑓 𝑚𝑜 < 2

26

𝐾𝑖 = 𝐶𝑖 = 1.0 𝑖𝑓 𝑚𝑜 > 2

Equation 14

where mo is the modified contact ratio, defined as [10],

𝑚𝑜 = 𝑚𝑝2 + 𝑚𝑓2

Equation 15

where mp is the transverse contact ratio and mf is the face contact ratio. Calculation of

these variables requires additional equations and is therefore limited to Appendix C.

The load sharing ratio, mn, is also dependent on the modified contact ratio, mo, and

determines what proportion of the load is carried on the most heavily loaded tooth. The

load sharing ratio is determined by [10],

𝑚𝑛 = 1.0 𝑖𝑓 𝑚𝑜 < 2

𝑚𝑛 =𝑚𝑜3

𝑚𝑜3 + 2 𝑚𝑜2 − 4 3 𝑖𝑓 𝑚𝑜 > 2

Equation 16

The remaining design factors consist of the reliability factor, the correlation

factor, the surface finish factor, and the size effect factor, all of which have an effect on

the endurance limit of the material. For transmission shafts made of steel, the design

standard is to use a reliability factor equal to 3σ, which equates to a value of .7 for the

reliability factor, Fr, and 1.0 for the correlation factor, K*. The surface finish factor, Kfs,

is determined based on the surface finish of the manufactured component and is used to

apply conservatism to the manufacturing processes that will be used. If the surface will

be ground, the value of Kfs is taken to be 1.0 but if the final component will not be

ground and instead, machined in some other manner, Kfs is taken to be 1.33 for steel with

an ultimate tensile strength equal to or greater than 200 ksi, or 1.25 for steel with an

ultimate tensile strength equal to 136 ksi.

The size effect factor, Fs, not to be confused with the size factor, Ks, is not as

simple however. Calculation of this value is based on a reduction to the mean endurance

limit due to the nature of geometrically similar parts decreasing with increasing size of

27

the part. This reduction is thought to be caused by the chain analogy which shows

statistically that the mean strength of a number of identical units in series decreases as

the number of units increase. The nature of this phenomenon is given by [10],

. 51

𝑉.𝑅. = 1

2𝜋𝑒−

𝑡2

2 𝑑𝑡

−∞

𝑡

Equation 17

where t is equal to the stress in numbers of standard deviations from the mean, and V.R.

is the volume ratio defined as the volume of the critically stressed area of the component

equal to that of the standard R-R Moore specimen which is used to develop the

allowable stresses upon which fatigue data is based. Numerous texts [6] give the value

of this integral as [10],

𝐹𝑠 = (1 − 𝜈)𝑡

Equation 18

where ν is equal to the coefficient of variation for the material. Mathematically, the

volume ratio for steel can be found using [10],

𝑉. 𝑅. =𝑉

. 009

Equation 19

with V equal to the critically stressed volume. This is also known as the volume of

stressed material of the component which is within one-third of the maximum stress of

the component. The step by step procedure for determining the size effect factor, Fs, is

shown in Section 3, Results and Discussion.

28

3. Results and Discussion

It has been shown thus far that the design and analysis of a bevel gear is heavily

dependent on mathematical equations. To keep track of the variables and iterative

procedures, three Microsoft Excel spreadsheets were generated, one for the analysis of

the gear shaft, shown in Appendix B, one for the calculation of the spiral bevel data and

stress values, shown in Appendix C, and one for the gear life calculation, shown in

Appendix D. The second spreadsheet utilizes the Macro feature in excel to calculate the

necessary terms. The macro feature also launches a Visual Basic computer program

which is used to perform the more complicated mathematical functions, like if-then

statements and Do loops. The details of the program will not be discussed, only the

results that have been generated. Also, this section will not cover the calculation of

every variable or equation as these can be seen in the attached appendices. Instead, a

broad scope of the analysis will be discussed to give the reader a general understanding

of the work that was performed and the results.

The analysis begins with calculation of the gear loads generated by the spiral

bevel mesh and reaction loads generated at the tapered roller bearings. Using Equation

2, Equation 4, and Equation 5, the tangential load, Wtp, axial thrust load, Wa, and

separating load, Wr, are calculated. The bearing reaction loads previously presented in

Figure 9 are calculated through application of Newton’s Second Law by summing the

forces in the axial plane and summing moments about points A and B, both in the

vertical and horizontal plane. Figure 10 below provides the necessary geometric

relationships necessary to perform the moment summations.

29

Figure 10 - Detailed location of loading

Solving all of the appropriate equations, details of which are shown in Appendix A,

gives the following values for the gear tooth loads and bearing reaction loads.

Table 4 - Calculated gear tooth loads and bearing reaction loads

Wtp 2,140 lbs

Wa 1,629 lbs

Wr -705 lbs

RAV 2,048 lbs

RAH 2,586 lbs

RAA 1,629 lbs

RBV 1,343 lbs

RBH 446 lbs

30

3.1 Fatigue Analysis

Fatigue analysis is performed at the most critical sections of the gear, those where

the wall thickness is the smallest and the loading is the highest. For this gear, two

critical sections have been identified and are shown below in Figure 11.

Figure 11 - Location of critical sections

Critical section A-A, which is primarily affected by the stress concentration occurring as

a result of the adjacent radius, will be investigated first. The dimensional limitations are

defined by the minimum outer diameter and the maximum inner diameter, which results

in the thinnest wall thickness. The minimum outer diameter is 1.940 inches and the

maximum inner diameter is 1.780 inches. The section modulus for the hollow

cylindrical section can now be calculated given the critical dimensions.

The bending moment at critical section A-A is a vectoral combination of two

planes and is calculated using the formula,

31

𝑀 = ( 𝑅𝐵𝑉 ∗ 𝑋4 2 + (𝑅𝐵𝐻 ∗ 𝑋4)2

Equation 20

where X4 is equal to the horizontal distance, also known as the moment arm, from

section A-A to the line of action through which the loads RBV and RBH act. Once the

moment load is determined, vibratory bending can be calculated using the moment load

and the section modulus as discussed above. Vibratory bending is defined as [10],

𝑓𝑣𝑖𝑏 =𝑀

𝑍

Equation 21

and is shown in the table below. Next, steady torsion is calculated using [10],

𝑓𝑠 =𝑇

2𝑍

Equation 22

Vibratory bending and steady torsion are then combined to calculate the principle steady

stress acting at section A-A. The principle steady stress is defined as [10],

𝑓𝑠𝑡𝑒𝑎𝑑𝑦 =𝑓𝑎2

+ 𝑓𝑎2

2

+ 𝑓𝑠2

Equation 23

where fa is the normal stress acting at section A-A. In this case, the normal stress is

equal to zero because there is no direct axial force acting in the plane. Therefore, fsteady

is equal to the steady torsion. The calculations are shown in detail in Appendix B and

summarized in the table below.

32

Table 5 - Calculated values at critical section A-A

Z 0.209 in3

M 2,705.5 in-lbs

fvib 12,945 psi

fs 10,366 psi

fsteady 10,366 psi

Using this steady stress, an equivalent vibratory stress can be found using Figure 12

below.

Figure 12 - Constant-life fatigue diagram for heat-treated AISI 4340 alloy steel, Ftu

= 150 ksi, Kt = 1.0 [10]

A steady stress value of 10,366 psi, as calculated above, results in a vibratory stress of

69,000 psi when using Figure 12 above. This figure is derived using a value of 150,000

psi for the ultimate tensile strength of 4340 steel. Similar data for SAE 9310 steel does

not exist and therefore a reduction factor will be applied. A core hardness value of

Rockwell hardness number C 30 – 45 for SAE 9310 steel results in an ultimate tensile

strength, Ftu, of 136,000 psi, not 150,000 psi. Therefore, an adjusted endurance limit,

Fen’, is calculated by applying the reduction factor,

33

𝐹𝑒𝑛 ′ = 69,000 ×136,000

150,000= 62,560 𝑝𝑠𝑖

Equation 24

This value for the endurance limit is modified further to account for additional design

parameters such as the size effect factor, correlation factor, surface finish factor, and

reliability factor, previously discussed in Section 2.7. This further modification is

performed using [10],

𝐹𝑒𝑛 = 𝐹𝑒𝑛 ′ 𝐹𝑠 𝐹𝑟 𝐾

∗

𝐾𝑓𝑠

Equation 25

Because the size effect factor was only briefly discuss in Section 2.7, a value still needs

to be determined for Kfs. To begin, the volume ratio of critically stressed material, Vcr, is

calculated at section A-A, where the minimum outer diameter is 1.940 inches and the

maximum inner diameter is 1.780 inches. Using the recommendations provided by

Figure 13 below,

Figure 13 - Volume of stressed material for shaft subjected to rotating bending [10]

34

and knowing that the design configuration can be described as a fillet where Di >

.67*Do, the following equation is used to calculate Vcr [10],

𝑉𝑐𝑟 =𝜋𝜌𝑟 𝐷𝑜

2 − 𝐷𝑖2

2

Equation 26

where ρr is equal to the minimum size of the fillet radius, which is .240 inches.

Therefore, Vcr is calculated to be .22 in3. Next, the volume ratio, V.R., is calculated

using Equation 19, which gives a V.R. value equal to 24.4 in3. Using Figure 14 below, a

size effect factor, Kfs, can be derived from the volume ratio, V.R.

Figure 14 - Size effect factor as a function of the volume ratio [10]

Following the curve for steel, where v equals .10, and finding the calculated V.R. of 24.4

in3 along the horizontal axis, gives a size effect factor, Fs, equal to .815. Revisiting

Equation 25, the modified endurance limit is calculated to be,

35

𝐹𝑒𝑛 = 62,560 . 815 . 70 1.0

1.25= 28,552 𝑝𝑠𝑖

Equation 27

Now that the endurance limit has been fully adjusted, the fatigue margin of safety, M.S.,

can be calculated using the formula [10],

𝑀. 𝑆. =𝐹𝑒𝑛

𝐾𝑡 × 𝑓𝑏− 1

Equation 28

Therefore, the fatigue margin of safety for section A-A is equal to .48. The positive

margin of safety means that during normal operating conditions critical section A-A will

not fail throughout the intended design life of 50,000 hours. Obviously, the margin of

safety can be increased but a fine line exists between robustness of the design and

weight. In helicopter applications, weight is a crucial factor. One of the ways the

margin of safety could be increased is to increase the wall thickness of section A-A, but

this would result in a heavier gear. Being that the margin of safety is already as high as

.48, there is no need to implement additional factors since the component will not fail

under normal conditions.

A very similar methodology is followed to investigate the second critical section

identified, section B-B, previously shown in Figure 11. The main difference in the

analysis at section B-B is that an axial load has to be accounted for as a result of the

axial loading from the locking nut that keeps the gear in place. The axial force is

actually a pre-load force based on the torque applied to the nut. The maximum torque

value to be applied to the nut is 125 ft-lbs which is equal to 1,500 in-lbs. Certain design

features of the locking nut must be known in order to calculate the torque coefficient of

the nut, K, which is defined as [6],

36

𝐾 =𝑙 + 𝜋𝜇𝑓𝑎𝑡 sec θ

2 𝜋𝑎𝑡 − 𝜇𝑓𝑙 sec 𝜃 +

𝜇𝑓𝑐

2𝑎𝑡

Equation 29

where,

Table 6 - Design properties of locking nut

number of thds per inch (N): 16 thd/in

lead (l): 0.0625 in/thd

thd pitch dia. (at): 1.2082 in

coefficient of friction (μf): 0.16

pressure flank angle (θ): 7 degrees

mean collar dia. Of nut (c): 1.510 in

Therefore, K is calculated to be .1891. Next, the torque applied to the nut can be

converted into an axial pre-load using the formula [10],

𝑃 =𝑇

𝐾𝑎𝑡

Equation 30

where T is the torque in inch-pounds and is 1,500 in-lbs as previously stated.

Substituting the known values, the axial preload, P, is equal to 6565.4 lbs.

The axial stress, defined as [10],

𝑓𝑎 =𝑃

𝐴𝑟

Equation 31

where Ar is equal to the cross-sectional area of section B-B and can be found using [10],

𝐴𝑟 =𝜋 𝑂. 𝐷.2− 𝐼. 𝐷.2

4

Equation 32

37

where O.D. and I.D. are the minimum outer diameter and the maximum inner diameter

respectively. This produces a cross-sectional area of .896 in2 and an axial stress, fa, of

7,327 psi.

Following the same methodology for section A-A that was previously discussed,

the table below was generated to show the calculated values for vibratory bending, fvib,

steady torsion, fs, and principle steady stress at critical section B-B. The table also

shows the result of using Figure 12 to convert the calculated steady stress to a vibratory

stress and the resulting initial endurance limit.

Table 7 - Calculated values at critical section B-B

Z 0.222 in3

M 695 in-lbs

fvib 3,131 psi

fs 9,759 psi

fsteady 14,087 psi

fv 68.5 ksi

Fen' 62,107 psi

Further reduction of the endurance limit is performed using the same values for the

correlation factor, surface finish factor, and reliability factor as those used in the analysis

performed on section A-A. The size effect factor needs to be recalculated however,

based on the maximum allowable dimension for the inner diameter of .905 inches and

the minimum allowable dimension for the outer diameter of 1.400 inches. In this case, a

different equation is used to calculate the volume of critically stressed material because

Di ≤ .67*Do. The proper equation is shown in Figure 13, and results in a V.R. value of

8.89 in3. By again employing Figure 14, a size effect factor, Fs, is determined to be .86.

Now that all of the design factors are known, further reduction of the endurance limit is

performed using Equation 25, resulting in a modified endurance limit of 29,911 psi.

Equation 28 is then utilized to find the margin of safety, M.S., for critical section B-B,

which results in a M.S. value equal to 3.34. A step by step procedure of the analysis at

section B-B is presented in Appendix B.

38

3.2 Static Analysis

In addition to fatigue analysis, a static analysis is conducted on the gear shaft in

order to account for any peak loads which may occur during operation. As previously

discussed, peak loads do occur and can be viewed in Appendix A. Federal aviation

requirements published by the Federal Aviation Administration, which governs the

design and operation of commercial aircraft throughout the United States of America,

establish design parameters that state that a static analysis must be conducted at twice

the normal operating condition. In this application, the design horsepower to which this

gear has been designed is 240HP which is the value at which the fatigue analysis was

conducted. For the static analysis conducted here, a value of 590HP will be used, which

far exceeds the FAA requirement of twice the normal operating condition.

The static analysis is conducted at the location of the shaft that experiences the

highest loading, which in this case occurs at section B-B due to the additional axial load

caused by the locking nut. A procedure similar to that used for the fatigue analysis is

performed with the replacement of the higher torque value, 590HP instead of 240HP.

As a result, the axial preload, bending moment, axial stress, vibratory bending, and

steady stress remain unchanged whereas the steady torsion increases to 23,991.7 psi

based on Equation 22. The static margin of safety, M.S., defined as [10],

𝑀. 𝑆. =1

1.5 𝑓𝑎𝑓𝑢𝑙𝑡

2

+ 4 𝑓𝑠𝑓𝑢𝑙𝑡

2

− 1

Equation 33

is then calculated with fult equal to the ultimate tensile strength of SAE 9310 steel, which

is 136 ksi. Substituting the known values results in a static margin of safety equal to .87.

A fully detailed approach to the static analysis is shown in Appendix B.

3.3 Calculation of Hertz Stresses (Pitting Resistance)

Once the design of the gear shaft has been verified through static and fatigue

analysis, focus shifts to conducting analysis on the gear teeth. First, calculation of the

39

Hertz stresses will be performed in order to gauge the ability of the gear teeth to resist

pitting. The methodology previously discussed in Section 2.5 and Section 2.7 will be

employed to calculate the geometry factor for pitting resistance, I, shown in Equation 12,

which will then be used to find the value for compressive stress acting on the gear teeth,

fc, shown in Equation 6.

Calculation of the geometry factor, I, is a complicated process that involves

solving ten equations iteratively. Before starting, the values for outer cone distance,

large end addendum, pitch diameter, net face width, number of teeth, diametral pitch,

pitch angles, face angles, normal pressure angle, and mean spiral angle must be known.

Equations for these values are presented in the AGMA standard, AGMA 2005-D03,

Design Manual for Bevel Gears, and a sample list of calculations is shown in Appendix

A of that document. Calculation of every required variable will not be discussed here,

but is shown in Appendix C for reference. Once these values are known, the remaining

variables can be calculated.

Now, to begin the iterative process, an assumption is made for f, the distance

from the mid-point of the tooth to the line of action. This value is then used to solve for

the length of line of contact, s, and the load sharing ratio, mN, used directly in the

calculation of the geometry factor. Using the assumed value for f, a typical iteration

follows the procedure below:

40

Figure 15 - Iterative procedure to calculate the load sharing ratio, mN [10]

Once mN and s are calculated, the geometry factor, I, can be calculated. This process is

then continued until the geometry factor is minimized.

Design experience led the iterative procedure to begin with a value for f equal to

1.0. This prevented calculation of the load sharing ratio and the geometry factor because

values for η1 and η2 are non-existent because the result is an imaginary number. The

table below shows the results of the iterations and the dashes represent iterations that

could not be finished, as described above. The iterative procedure was continued until

the geometry factor was successfully minimized.

41

Table 8 - Results for calculating the load sharing ratio and geometry factor

f 1.0 0.5 0.25 0.15 0.10 0.05 0.01 0.002

mn - - 0.335146 0.60129 0.739121 0.884239 0.985792 0.997657

I - - 0.2450 0.1658 0.1410 0.1206 0.1086 0.1072

Further reduction of f has very little effect on the value for I, which shows that the

minimization procedure has been successfully completed. In addition, a load sharing

ratio value, mN, close to 1.0 is definitely sufficient. An mN value of exactly 1.0 would

mean that the pinion and gear share the applied load equally, which is an ideal case and

is not typical in most applications. The fact that an mN value close to 1.0 has been

achieved is a marked example of the accuracy of this iterative procedure.

Once the geometry factor was calculated, the remaining items required to

calculate the Hertz stresses could also be finalized. Cp was calculated using Equation 7,

Wt using Equation 2, and the remaining design factors were chosen based on previous

discussion in Section 2.7. Combining all of these known values, the Hertz stresses in the

gear teeth were calculated to be,

𝑓𝑐 = 180.6 𝑘𝑠𝑖

Equation 34

3.4 Calculation of Bending Stresses

Analysis now shifts to establishing the value for bending stresses in the gear teeth

based on the geometry of the teeth and the applied loads. Calculation of the bending

stresses includes utilizing the size factor, Ks, the load distribution factor, Km, and the

geometry factor, J, previously discussed in Sections 2.5 and 2.7. Using Equation 11, a

value for the size factor, Ks, was calculated to be .660 and the load distribution factor,

Km, was assumed to be 1.10 as previously explained. The remaining calculation to

perform prior to calculating the bending stress is the calculation of the geometry factor,

J, using Equation 13. The only remaining variable needed is Yk, the tooth form factor,

which incorporates both the radial and tangential components of the normal load applied

to the gear teeth. The tooth form factor is calculated using [10],

42

𝑌𝐾 =2

3

𝑃𝐼𝑇𝐶𝐻

𝑘𝑓 1𝑋𝑛

−tan 𝜙𝑛

3𝑡𝑛

Equation 35

where kf is the actual stress concentration factor, derived from the theoretical stress

concentration factor, kt, Φn is the angle which the normal force makes with a line

perpendicular to the tooth centerline, Xn is a ratio which defines the gear tooth strength

factor, and tn is one-half the tooth thickness at the critical section of the gear tooth.

Calculation of Xn, the gear tooth strength factor, involves an iterative process to

accurately define this ratio. The iteration process appears below,

Figure 16 - Iterative procedure to calculate tooth form factor, Xn [10]

with a value of θ less than .5 as a recommended initial value. In this application

however, because of the limitations of the formulas above, a value for the ratio does not

43

turn positive until θ is equal to .1, but this result produces a ratio value of only .11695,

not close to the desired value of .5. Continual reduction of θ leads to a range for θ

between .05 and .06. Table 9 below displays computed results using assumed values for

θ in the range of .05 to .06, and the effect this has on the important factors in the bending

stress analysis.

Table 9 - Assumed values for θ and its effect on bending stress

θ 0.05 0.0575 0.059 0.05935 0.06

Xn 0.82642 0.54582 0.50825 0.50007 0.48541

Yk 0.35877 0.35268 0.35237 0.35233 0.35228

J 0.2489 0.24275 0.24217 0.24205 0.24186

Bending Stress (ksi) 30.6 31.4 31.5 31.5 31.5

To complete Table 9, a value for θ was assumed and the remaining values for Xn, Yk, J,

and Bending Stress were calculated based on the assumed θ values. Because the

ultimate goal in this analytical procedure is not to reduce the bending stress but to

accurately calculate the bending stress, a θ value of .05935 will be used because it

produces the closest Xn value to the desired value of .5, even though it results in a higher

bending stress. The higher bending stress however, 31.5 ksi compared to 30.6 ksi, is

only an increase of .9 ksi, or 900 psi, which is only 2.9% of the applied bending stress

and is therefore considered minimal. Therefore, a J value of .24205 will be substituted

into Equation 8 previously discussed, in order to calculate the bending stress in the gear

teeth. This results in a value for fb of,

𝑓𝑏 = 31.5 𝑘𝑠𝑖

Equation 36

Now that the analytical procedures have been performed in order to identify the

bending stresses and compressive stresses in the gear teeth, the next step is to compare

the calculated values to the allowable values to ensure that the design is safe for

operation under the specified parameters. The data from Table 10 below was compiled

from Tables 3 and 5 in the AGMA 2003-B97 standard, Rating the Pitting Resistance and

Bending Strength of Generated Straight Bevel, Zerol Bevel, and Spiral Bevel Gear

44

Teeth. This extracted data represents the results of laboratory and field experience for

the specified material and condition of that material, and is greatly dependent on

material composition, cleanliness, quality, heat treatment, mechanical properties, forging

practices, residual stress, final processing operations in manufacture and method of

stress calculation [8]. Discussion of these factors is covered in the AGMA standard and

should be adhered to in order for these values to accurately represent the design and

analysis covered throughout this paper. Because the design and analysis presented has

followed the recommendations of the AGMA standard, the allowable stress values

presented below are accurate to this type of application.

Table 10 - Allowable stress values [8]

Material

Designation Heat Treatment

Minimum Surface

Hardness

Compressive Stress Allowable (ksi)

Grade 1 Grade 2 Grade 3

Steel

Carburized & Case

Hardened 58 – 64 HRC 200 225 250

Material

Designation Heat Treatment

Minimum Surface

Hardness

Bending Stress Allowable (ksi)

Grade 1 Grade 2 Grade 3

Steel

Carburized & Case

Hardened 58 – 64 HRC 30 35 40

The appropriate Grade, as specified above, is chosen based on Table 3 from the AGMA

standard which provides hardness recommendations for the core of the gear as well as

the gear teeth. For Grade 3 gears, Table 5 from the AGMA standard recommends a

hardness value of 58 – 64 HRC for the gear teeth and a hardness value of 30 HRC

minimum for the core (center of tooth at root diameter). Because these are the hardness

values that have been chosen for this specific gear application, the values for a Grade 3

gear are the true allowable values to which the calculated values should be compared.

As such, an allowable value for compressive stress is 250 ksi which is much higher than

the calculated 180.6 ksi compressive stress for this application. The allowable value for

bending stress, 40 ksi, is also much higher than the calculated value of 31.5 ksi. This

45

analysis proves that the finalized gear tooth design has been shown to be safe for

operation in this specific application.

3.5 Gear Life Calculations

Gear life calculations were performed in accord with AGMA recommendations,

specifically utilizing the Miner’s rule methodology presented earlier in Section 2.6. The

flight spectrum, presented in Appendix A, shows five flight maneuvers in which fatigue

damage occurs to the gear teeth. Using a Microsoft Excel spreadsheet specifically

designed for the calculation of damage accumulation, the five maneuvers were input into

Appendix D, along with the composite percent time and input power associated with

each of the five maneuvers. The input power is then converted to a bending stress and a

compressive stress, shown in columns F and J respectively, and compared to the

allowable stresses in accord with AGMA recommendations shown above in Table 10.

Using the calculated stresses and Equation 10 previously discussed, individual damage

occurrences are calculated for each maneuver and then summed in order to obtain a life

calculation for both bending and durability. To perform these calculations, IF-THEN-

ELSE statements were used to determine the true extent of each damage occurrence.

The calculated fatigue life was then compared to the required 50,000 flight hours. If the

calculated fatigue life is greater than 50,000 hours, the gear is said to have unlimited life

for application in this helicopter. As Appendix E shows, the gear designed and analyzed

herein has unlimited life for both bending life and durability life.

46

4. Conclusion

A spiral bevel gear has been designed and analyzed using current industry

standards combined with the implementation of learned methodology through years of

design experience and test results. The gear was designed for use in an intermediate

gearbox of a medium class helicopter and was framed around the existing transmission

components in use, specifically utilizing the current transmission housings, bearings, and

seals. A detailed summary of material selection, material processing, design of gear

teeth, and selection of design factors was presented in order to clarify the proper

selection of certain design parameters. Upon completion of the design phase of the gear,

analysis was conducted to ensure appropriate margins of safety had been implemented

into the design.

Gear loads were calculated based on geometry of the spiral bevel gear teeth and

bearing support structure. Fatigue analysis was then conducted at the most critical

sections of the gear. Margins of safety were calculated at the two critical sections and a

margin of safety equal to .48 was determined at section A-A, shown in Figure 11. A

margin of safety equal to 3.35 was determined at section B-B, also shown in Figure 11.

Static analysis was then performed at section B-B, the highest loaded section of the gear

shaft. The static analysis was conducted at approximately 2.5 times the endurance limit

of the gear, exceeding the Federal Aviation Administration recommendation of 2.0 times

the endurance limit. This static analysis at section B-B produced a margin of safety

equal to .87. A positive margin of safety was shown to provide adequate safety for

operation in this application.

Upon completion of the analysis of the gear shaft, the analytical focus shifted to

the gear teeth. Geometry factors for pitting resistance and bending strength were

calculated using iterative procedures that were explained in detail. Hertz (compressive)

and bending stresses in the gear teeth were calculated using the recommended practices

of the American Gear Manufacturing Association (AGMA). The Hertz stresses were

calculated to be 180.6 ksi and the bending stresses were calculated to be 31.5 ksi. Per

the AGMA standards, allowable stresses in carburized and case hardened gear teeth are

47

250 ksi and 40 ksi respectively. As a result, the stresses produced in the gear teeth were

acceptable, mitigating the risk of failure to the designed gear teeth.

Finally, fatigue life calculations were performed using the estimated flight load

spectrum and the specific flight maneuvers that cause fatigue damage to the gear teeth.

Miner’s rule was explained and utilized to perform the necessary fatigue life

calculations, which resulted in unlimited life for the gear under the specified design

parameters.

48

5. References

[1] Askeland, Donald R., and Pradeep P. Fulay. The Science and Engineering of

Materials. New York: Cengage Engineering, 2005.

[2] Callister, William D. Jr. Materials Science and Engineering An Introduction. New

York: John Wiley & Sons Inc., 2003.

[3] “Design Manual for Bevel Gears.” ANSI/AGMA 2005-D03 (2003).

[4] Dieter, George E. Mechanical Metallurgy. Boston, MA: McGraw-Hill, 1986.

[5] “Gear Materials, Heat Treatment and Processing Manual.” ANSI/AGMA 2004-C08

(2007).

[6] Horton, Holbrook L., Franklin D. Jones, Erik Oberg, and Henry H. Ryffel.

Machinery’s Handbook, 28th

Edition. New York: Industrial Press, 2008.

[7] Mott, Robert L. Machine Elements in Mechanical Design. New Jersey: Pearson

Education Inc., 2004.

[8] “Rating the Pitting Resistance and Bending Strength of Generated Straight Bevel,

Zerol Bevel and Spiral Bevel Gear Teeth.” ANSI/AGMA 2003-B97 (1997).

[9] United Technologies Corporation. Sikorsky Structures Manual. Connecticut:

Sikorsky Aircraft Corporation, 1992.

[10] United Technologies Corporation. Transmissions Design Manual. Connecticut:

Sikorsky Aircraft Corporation, 1990.

49

6. Appendix A

Helicopter flight spectrum with anticipated horsepower and torque loads. See Microsoft

Excel file titled, “Flight Spectrum and Anticipated Load Conditions” on the associated

CD.

50

7. Appendix B

Fatigue and static analysis on gear shaft. See Microsoft Excel file titled, “Fatigue and

Static Analysis” on the associated CD.

51

8. Appendix C

Calculation of geometry factors, compressive stresses and bending stresses. See the

Microsoft Excel file titled, “Spiral Bevel Gear Data” on the associated CD.

52

9. Appendix D

Fatigue life calculations. See the Microsoft Excel file titled, “Spiral Bevel Gear Life

Calculation” on the associated CD.