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
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
6. Appendix A .................................................................. .............................................. 49 7. Appendix B .................................................................. .............................................. 50 8. Appendix C .................................................................. .............................................. 51 9. Appendix D .................................................................. .............................................. 52
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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 . and its effect on bending stress ................ ..................... 43 Table 10 - Allowable stress values [8] ......................................... ................................... 44
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
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 -
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 -
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
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 ag Gear addendum angle deg ap Pinion addendum angle
deg G Gear pitch angle deg GR Gear root angle deg Go Gear face angle of blank deg . Pinion pitch angle deg .o Pinion face angle of blank deg .R Pinion root angle deg dg Gear dedendum angle deg dp Pinion dedendum angle deg .FH' Heel increment .FT' Toe increment
. Pressure flank angle deg µ Poisson's ratio µf Coefficient of friction . Coefficient of variation p Pi . Profile radius of curvature at pitch circular in mean normal section in .r Minimum fillet radius in .o Relative radius of curvature in S Shaft angle deg . Pressure angle
deg .h Pressure angle at point of load application deg .n Angle at which the normal force makes with a line perpendicular to the tooth centerline deg . Mean spiral angle deg .b Base spiral angle deg
ACKNOWLEDGMENT First and foremost, the author wishes to thank his family and friends who have s upported 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 Aircraf t who were always willing to share their extensive knowledge of gear design and analys is.
ABSTRACT This investigation gives a detailed approach to spiral bevel gear design and analysis. Key design parameters are investigated in accord with industry standar ds and recommended practices for use in a medium class helicopter. Potential gear mater ials 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 g ear 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 s ections of the gear shaft resulting in margins of safety equal to .48 and 3.35. Static anal ysis is conducted at the most critical section in accordance with Federal Aviation Administration requirements, resulting in a margin of safety equal to .87. Furth er 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 k si which allows for proper resistance to pitting. Bending stresses are calculated equal t o 31.5 ksi which shows proper bending strength in the gear teeth to mitigate the risk of fa ilure to a gear tooth. Results are compared to the recommended allowable stresses as publis hed by the American Gear Manufacturing Association. Finally, fatigue life calculatio ns are performed to show that the gear has been designed with unlimited life for this s pecific application.
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 d ifferent direction. Throughout the mechanical industry, many types of gears exist with ea ch 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 fo rmed during gear mesh which allows contact to begin at one end of the tooth (toe) and smooth ly progress to the other end of the tooth (heel), as shown below in Figure 1. Spira l bevel gears, in comparison to straight or zerol bevel gears, have additional overlappi ng tooth action which creates a smoother gear mesh. This smooth transmission of power alo ng 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 reduct ion, make them a prime candidate for use in the helicopter industry.
Heel Heel Toe Toe Figure 1 - Spiral bevel gear mesh [3]
The American Gear Manufacturing Association (AGMA) has developed standards for the design, analysis, and manufacture of bevel gears. The first st ep in any general design employing gears is to first predict and understand all of the con ditions under which the gears will operate. Most importantly are the anticipated loads a nd speeds which will affect the design of the gear. Additional concerns are the ope rating 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 upgrad e program for an existing helicopter firmly established in the medium class commer cial 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 spe ed 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 he licopter that operates more smoothly than previous models. It is important to note that furthe r 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 chan ging 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 a t 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 pound s when filled with approximately .260 gallons of oil, which lubricates the gears by spl ash 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 rotat e out of mesh, they dip through the oil and lubricate the gear teeth prior to meshing. Th is oil film prevents scuffing and scoring of the gear teeth and helps reduce the friction an d heat generation caused by the clash of the gear teeth. Centrifugal forces, an outward force
C:\Users\Matt\Desktop\RPI\Engineering Project\2D Cutaway - IGB Input Assy with L abels.bmp associated with rotation, also play a big part in splash lubricating the gear me sh. As the gears rotate through the oil, the centrifugal force flings the oil against the w alls of the center transmission housing. This housing was designed with a reservoir at the t op 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 th rough two drip ports to lubricate the outer bearing of both the input and output assemblie s. The configuration of the input assembly of the intermediate gearbox can be seen in F igure 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 th e
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 millio n 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 th an ten million cycles, a value equal to one half the value of the peak load or the high est 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 theref ore introduce a conservative approach into the design effort. Fortunately, in this a pplication, the full flight spectrum is known because of the long service history of this he licopter. A wide ranging flight spectrum has been established that displays the changing spe eds, 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 fligh t spectrum that is anticipated for the upgraded model of the helicopter. The estim ated flight spectrum is based on analytical tools proprietary to Sikorsky Aircraft an d therefore will not be discussed here in detail. It does however estimate performance param eters of the helicopter and the loads that will occur in flight based on the overall desi gn 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 v alues are converted to horsepower. This can be done using the formula [7], ....= ..*...... 5252 Equation 1
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 Reg ime #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 conditio n 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 t hat 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 ulti mate 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 conser vatism into the design of the gear.
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 cons idered are through hardening and surface hardening, which includes but is not limited t o carburization, nitriding, induction hardening, and flame hardening. Through hard ened steels are used when medium wear resistance and load carrying capacity are desir ed whereas carburized and hardened gears are used when high wear resistance and hig h load carrying capacity are required [5]. Specifically, the desired loading and d esired design life are integral in selecting the proper material and any additional tre atment 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 t hese 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 o nly those which are easily carburized and case-hardened. Table 1 below, generated from AGM A recommendations of associated steel grades and their typical heat treatments, di splays 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
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 t he case hardenability for the alloy steels shown in Table 1 and Table 2.
Figure 3 - Case hardenability of carburizing grades of steel [5]
The horizontal axis of Figure 3 is the ruling section, also called the controlli ng 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. Usi ng 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 sec tion. 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 manufact ured. 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, a nd the chemical compound iron carbide in a form metallurgically known as cementite. An important third microconstituent is a microcomposite consisting of cementite pla telets 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 pr esent, called bainite, another mixture of carbide and ferrite. When carbon steel of thi s 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 a nd cementite will be fully converted to austenite. This can be seen in the phase diagram of c arbon steel, shown below in Figure 4.
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 gre atly 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 ra te, will cause the austenitized steel to form a new structure called martensite. This microstructure is characterized by an angular needlelike structure and a very hi gh hardness, thereby making it highly desirable in applications where high wear res istance and load carrying capacity are required. Figure 5 below illustrates the differen ce in size and shape of the microstructures of pearlite, bainite, and martensite at a magni fication of x7500.
Figure 5 - Electron micrographs of (a) pearlite, (b) bainite and (c) martensite (x7500) [1] More specifically, the relationship between a steels mechanical properties and t he 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 mea sures 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 rep lace pearlite and bainite during the cooling process. Figure 6 below displays a harde nability curve for several alloy steels.
Figure 6 - Hardenability curves for several steels [1] Using an industry standard, the Jominy distance test, Figure 6 shows that hardne ss decreases with increasing distance from the quench surface. Therefore, steel reg arded as highly hardenable will retain large values of hardness for relatively long dista nces. The important feature of the figure above is the varying curve shapes displayed by t he alloy steels. The 1050 and 4320 alloys show a shallow depth of hardness below the surf ace whereas the other four alloys exhibit a high hardness persisting to a much great er 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 d isparity 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-t opearlite and/or bainite reactions which permits more martensite to form for a pa rticular cooling rate, yielding a greater hardness [2]. The maximum attainable hardness o f any steel is only realized when the cooling rate in quenching is rapid enough to ens ure full transformation to martensite [6].
In addition to the hardenability characteristics discussed above, the cooling ra te of a specimen can greatly affect the resulting hardness. Because the cooling rat e 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 result ing hardness. Of the three most popular quenching mediums, water, oil and air, oil i s the most suitable for heat treatment of most alloy steels as water is often too seve re and results in cracking or warping of a specimen. Air quenching often results in a p earlitic structure and is therefore ineffective in obtaining the desired martensitic stru cture. Geometry also affects the rate at which heat energy is dissipated to the quenchi ng 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 cool ing 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 struct ure like martensite. This process is typically performed using two essential steps; heati ng the steel to some temperature above its transformation point such that it becomes en tirely austenitic in structure, and then quenching the steel at some rate faster than t he critical rate in order to produce a martensitic structure. The resulting martensitic stru cture is mainly dependent on three factors (1) the composition of the alloy (austenite gr ain size and prior microstructure), (2) the type and character of the quenching medium (t ime 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 m ost 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 wi th 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 fur ther hardened at its surface by impregnating a components outer surface with a suffici ent
amount of carbon. This process, termed carburization, is a solution to applicati ons 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 tr eated in order to obtain a hard outer case and, at the same time, give the core the requi red physical properties. The term case hardening is ordinarily used to indicate the complete process of carburizing and hardening [6]. This paper will focus on carb urizing because that treatment has been chosen to be best suited for this gear applicati on. During the carburizing process, carbon is diffused into the parts 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 mol ten barium cyanide or sodium cyanide; gas carburizing, which involves heating the st eel in a gas of controlled carbon content; and pack carburizing, which involves sealing b oth the steel and solid carbonaceous material in a gas-tight container, then heating thi s combination [6]. The case depth, or resulting depth of carburization, is depende nt 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 degree s Farenheit with the temperature adjusted to obtain specific case depths for the i ntended application. Carburizing the entire part is typically not necessary, as it is on ly 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 quenching bath, such as a heat treated steel, often becomes desired internal strains. The brittleness and internal strains must machining so as to avoid fast fracture to the work piece. In order to ss and
sudden cooling in a brittle and forms un be removed prior to remove the brittlene
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 te mpering 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 ma y also be used to alter toughness, to refine crystal structure and grain orientation, o r to relieve stress or hardness from a working surface, all of which allow the gear to be mor e 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 as pects of bevel gear tooth design, starting from preliminary design values and progress ing towards a finished design ready to be analyzed. Not only does it give recommende d practices for design, it also covers manufacturing considerations, inspection me thods, lubrication, mounting methods, and appropriate drawing formats. While this is ce rtainly 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 pa rameters that should be used for one industry versus another. For example, the automotive industry typically uses cast iron gears in transmission applications whereas a c ast iron gear would not be feasible in a helicopter transmission because of the high leve l 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 intellige nt 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 appli cation, the bevel gears mesh at a shaft angle of 57 degrees.
Spiral angle and pressure angle are two design parameters that help determine th e 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 thi rty five degrees should be used. Following this common practice for selection of spiral a ngle establishes a good face contact ratio which maximizes smoothness and quietness d uring gear mesh. In regards to the selection of a pressure angle, a lower pressure ang le increases the transverse contact ratio, a benefit which results in increased ben ding strength, while also increasing the risk of undercut which is a major concern. L ower 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 becaus e the increased slot widths allow the use of larger fillet radii, resulting in increas ed bending strength. The contact stress is reduced however, as a result of the larger fille t radii, so close consideration is required to ensure the correct pressure angle is chosen f or the intended application. In addition, spiral bevel gears are designed such that the axial thrust load ten ds to move the pinion out of mesh. This helps to avoid the loss of backlash, defined a s the clearance between mating components. While a lot of backlash is not desirable, s mall 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 betw een 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 o f this gear was simplified because not everything had to be developed from scratch. Importan t geometric design parameters remained constant between the old gear and the new g ear 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 dia meter, pitch angles, and face width, which are the basis for calculating the necessary geometric
design parameters shown in Appendix C. These parameters are shown in Figure 7 an d Figure 8 below.
Figure 7 - Bevel gear nomenclature in the axial plane [3]
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 a ct as point loads applied at the mid-point of the face width of the gear tooth. The radial a nd 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 a re defined as [10], ......= 2.... ....-..sin.. Equation 2 for the pinion and,
......= 2.... ....-..sinG Equation 3 for the gear, with T equal to the torque, dp equal to the pitch diameter, F equa l to the face width, . equal to the pitch angle of the pinion , and G 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 axi al thrust load for a driving member (pinion) is defined as [10], ....= .... cos.. tan..sin..+sin..cos.. Equation 4 and the separating load is defined as, ....= .... cos.. tan..cos..-sin..sin.. Equation 5 where F is the pressure angle, and . is the mean spiral angle. Figure 9 below di splays the line of action through which the tangential, axial and separating loads act.
Figure 9 - Loads acting on gear The loads in Figure 9 above, labeled RBA, RBH, RBV, RAA, RAV, and RAH, are react ion 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 generate d 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 too th deterioration such as scuffing, wear, scoring, and case crushing are of less imp ortance, but proper design techniques established to employ designs for pitting resistanc e 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 fa ilure mode
of fatigue on the surface of the gear teeth under the influence of the contact s tress 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 Hert zian theory of the contact pressure between two curved surfaces and load sharing betw een adjacent gear teeth as well as load concentration that may result from uncertain ties in the manufacturing process. The contact stress is mainly a function of the square roo t of the applied tooth load. Three primary types of pitting are widely recognized through out the industry; they are initial pitting, micropitting, and progressive pitting. Initi al 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 pit s 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 remov ing 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 na ked 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 c reates large surface pits to start and progresses until a considerable portion of the tooth s urface has developed pitting craters of various shapes and sizes. The dedendum section of t he drive gear (pinion), shown previously in Figure 8, is often the first to experience se rious pitting damage and could potentially be the point of initiation of a bending fat igue crack, causing a tooth breakage failure [8]. The basic equation for compressive stress in a bevel gear tooth is given by [10] , ....=.... ....×.... .... 1 ..×.... .... ..
Equation 6
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], ....= 34.. .. (1-..2) Equation 7 where E is the Youngs modulus of the material and µ is Poissons ratio. For almost al l steels, including SAE 9310 the type used for this gear application, the values f or 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 var ious factors including: the compressive stresses at the tooth roots caused by the rad ial component of the tooth load; the non-uniform moment distribution of the load res ulting from the inclined contact lines on the teeth of spiral bevel gears; stress conce ntration 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], ....=.... .......... .. ....×.... .. Equation 8 where Wt is the tangential tooth load previously discussed, PITCH is the diametr al 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],
..........= .... .... Equation 9 where Np is the number of teeth on the pinion. 2.6 Gear Life Calculations Per the recommendations of the AGMA, Miners rule is used to calculate the effects of cumulative fatigue damage under repeated and variable intensity loads . Miners rule is based on the theory that the portion of useful fatigue life used u p by a number of repeated stress cycles at a particular stress is proportional to the t otal number of cycles in the overall fatigue life of the part. Using this hypothesis, Miners rule assumes that the damage done by each stress repetition at a given stress level i s 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 Miners rule, loads in excess of the gears endurance limit will cause damage. Failure is to be expected when [8], .... ...... .. ..=1=1.0 Equation 10 where k is the total number of different stress levels, ni is the number of cycl es at the i-th stress level, and Nfi is the total number of cycles to failure at the i-th stres s level. A detailed review of the load spectrum, presented in Appendix A, shows that a tota l 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 accumulatio n using Miners rule. Both bending life damage and durability life damage are calculated t o
ensure an adequate fatigue life for bending strength and pitting resistance. Det ails 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 lo ad conditions. It is typically influenced by design effects, manufacturing effects, transmission error, dynamic response, and resonance. In a broader sense, the dyn amic factor makes allowance for high-accuracy gearing which requires less derating th an lowaccuracy gearing and, at the same time, makes allowance for heavily loaded geari ng which requires less derating than lightly loaded gearing [8]. When gearing is manufactured using very strict processes and controls, resulting in very accurat e gearing, typical values of Cv between 1.0 and 1.1 are used. For this application, a Cv va lue 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 t hrough critical speeds, overspeed conditions, and braking (application of the rotor bra ke). 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 experie nced. Table 3 - Overload factors [8]
As previously discussed in Section 1, and shown in Appendix A, all peak loads an d 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 p eak 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 toot h manufacturing accuracy, tooth contact and spacing, alignment of the gear in its mounting, bearing clearances, and geometric characteristics of the gear teeth, a nd 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 bearin gs, the mounting of the gear is considered rigid which minimizes misalignment between th e gear, the bearings, and their mountings. Misalignment will exist however, as a r esult of assembly tolerances, backlash, and manufacturing tolerances. Design experience l eads 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 a nd 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 si ze to diameter of the part. The size factor can be quickly calculated using [10], ....= 1 ...........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 surfa ces where the calculated contact pressure will reach its maximum value [8]. The geometry f actor may be calculated from [10],
..= .. .... .. .. .... .. cos.. .... cos.. .... Equation 12 where A is the mean cone distance, Ao is the outer cone distance, s is the lengt h of line of contact, F is the actual face width, .o is the relative radius of curvature, Ki is the inertia factor, and mn is the load sharing ratio. Calculation of the geometry factor inc ludes an iterative process in order to minimize the distance from the mid-point of the to oth to the line of action because ultimately, it is desired to have the line of action go t hrough 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 considerati on 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, a nd 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 geome try factor [10], ..= .. .... .... .. .... .. .... ....×.... Equation 13 where Rt is the mean transverse radius of load application, R is the mean transv erse 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, c an be determined from [10], ....=....= 2.0 .... .... ....<2
....=....=1.0 .... ....>2 Equation 14 where mo is the modified contact ratio, defined as [10], ....= ....2+....2 Equation 15 where mp is the transverse contact ratio and mf is the face contact ratio. Calcu lation of these variables requires additional equations and is therefore limited to Append ix 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 too th. The load sharing ratio is determined by [10], ....=1.0 .... ....<2 ....= ....3 ....3+2 ....2-4 3 .... ....>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 3s, which equates to a value of .7 for the reliability factor, Fr, and 1.0 for the correlation factor, K*. The surface fini sh factor, Kfs, is determined based on the surface finish of the manufactured component and is u sed to apply conservatism to the manufacturing processes that will be used. If the surf ace will be ground, the value of Kfs is taken to be 1.0 but if the final component will n ot be ground and instead, machined in some other manner, Kfs is taken to be 1.33 for s teel 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 en durance limit due to the nature of geometrically similar parts decreasing with increasin g size of
the part. This reduction is thought to be caused by the chain analogy which show s statistically that the mean strength of a number of identical units in series de creases as the number of units increase. The nature of this phenomenon is given by [10], .51 ......= 12.. ....22.... -8 .. 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], ....=(1-..).. Equation 18 where . is equal to the coefficient of variation for the material. Mathematicall y, the volume ratio for steel can be found using [10], ......= .. .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 stre ss of the component. The step by step procedure for determining the size effect factor , Fs, is shown in Section 3, Results and Discussion.
3. Results and Discussion It has been shown thus far that the design and analysis of a bevel gear is heavi ly dependent on mathematical equations. To keep track of the variables and iterativ e procedures, three Microsoft Excel spreadsheets were generated, one for the analy sis 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 calcul ate 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-th en 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 calculat ion of every variable or equation as these can be seen in the attached appendices. Inst ead, a broad scope of the analysis will be discussed to give the reader a general under standing 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 Eq uation 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 prese nted in Figure 9 are calculated through application of Newtons 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.
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
3.1 Fatigue Analysis Fatigue analysis is performed at the most critical sections of the gear, those w here 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 oc curring as a result of the adjacent radius, will be investigated first. The dimensional lim itations are defined by the minimum outer diameter and the maximum inner diameter, which resu lts in the thinnest wall thickness. The minimum outer diameter is 1.940 inches and t he 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,
..= ( ......*..4 2+(......*..4)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] , ........= .. .. Equation 21 and is shown in the table below. Next, steady torsion is calculated using [10], ....= .. 2.. Equation 22 Vibratory bending and steady torsion are then combined to calculate the principl e steady stress acting at section A-A. The principle steady stress is defined as [10], ..............= .... 2+ .... 2 2+.... 2 Equation 23 where fa is the normal stress acting at section A-A. In this case, the normal st ress is equal to zero because there is no direct axial force acting in the plane. Theref ore, fsteady is equal to the steady torsion. The calculations are shown in detail in Appendix B and summarized in the table below.
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 Figu re 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 1 50,000 psi for the ultimate tensile strength of 4340 steel. Similar data for SAE 9310 s teel does not exist and therefore a reduction factor will be applied. A core hardness valu e of Rockwell hardness number C 30 45 for SAE 9310 steel results in an ultimate tensi le strength, Ftu, of 136,000 psi, not 150,000 psi. Therefore, an adjusted endurance limit, Fen, is calculated by applying the reduction factor,
......'=69,000×136,000150,000=62,560 ...... 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 fa ctor, and reliability factor, previously discussed in Section 2.7. This further modificati on is performed using [10], ......= ......' .... .... ..* ...... 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 mate rial, 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]
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], ......= ...... .... 2-.... 2 2 Equation 26 where .r 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 cal culated using Equation 19, which gives a V.R. value equal to 24.4 in3. Using Figure 14 b elow, 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. Re visiting Equation 25, the modified endurance limit is calculated to be,
......= 62,560 .815 .70 1.0 1.25=28,552 ...... Equation 27 Now that the endurance limit has been fully adjusted, the fatigue margin of safe ty, M.S., can be calculated using the formula [10], ......= ...... ....×.... -1 Equation 28 Therefore, the fatigue margin of safety for section A-A is equal to .48. The pos itive 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 mar gin 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 n ot fail under normal conditions. A very similar methodology is followed to investigate the second critical secti on identified, section B-B, previously shown in Figure 11. The main difference in t he analysis at section B-B is that an axial load has to be accounted for as a resul t 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 to rque value to be applied to the nut is 125 ft-lbs which is equal to 1,500 in-lbs. Cer tain design features of the locking nut must be known in order to calculate the torque coeff icient of the nut, K, which is defined as [6],
..= ..+..........sec.2 ......-......sec.. + ...... 2.... 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 (.): 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], ..= .. ......
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], ....= .. .... Equation 31 where Ar is equal to the cross-sectional area of section B-B and can be found us ing [10], ....= .. ......2-......2 4 Equation 32
where O.D. and I.D. are the minimum outer diameter and the maximum inner diamete r respectively. This produces a cross-sectional area of .896 in2 and an axial stre ss, 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 bendin g, fvib, steady torsion, fs, and principle steady stress at critical section B-B. The tab le 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 howeve r, based on the maximum allowable dimension for the inner diameter of .905 inches a nd 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 materi al because Di = .67*Do. The proper equation is shown in Figure 13, and results in a V.R. va lue 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 p si. Equation 28 is then utilized to find the margin of safety, M.S., for critical se ction B-B, which results in a M.S. value equal to 3.34. A step by step procedure of the ana lysis at section B-B is presented in Appendix B.
3.2 Static Analysis In addition to fatigue analysis, a static analysis is conducted on the gear shaf t in order to account for any peak loads which may occur during operation. As previou sly 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 Amer ica, 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 wh ich 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 anal ysis is performed with the replacement of the higher torque value, 590HP instead of 240H P. 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], ......= 11.5 .... ........ 2+4 .... ........ 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 e qual 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, calculat ion of the
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 wil l be employed to calculate the geometry factor for pitting resistance, I, shown in Eq uation 12, which will then be used to find the value for compressive stress acting on the g ear 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 di stance, large end addendum, pitch diameter, net face width, number of teeth, diametral p itch, 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 App endix 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 remai ning 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 t o 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 ite ration follows the procedure below:
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 pr ocess 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 facto r because values for .1 and .2 are non-existent because the result is an imaginary number. The table below shows the results of the iterations and the dashes represent iterati ons that could not be finished, as described above. The iterative procedure was continued until the geometry factor was successfully minimized.
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 th at the minimization procedure has been successfully completed. In addition, a load shar ing 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 Eq uation 7, Wt using Equation 2, and the remaining design factors were chosen based on previ ous discussion in Section 2.7. Combining all of these known values, the Hertz stress es in the gear teeth were calculated to be, ....=180.6 ...... 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 ben ding stresses includes utilizing the size factor, Ks, the load distribution factor, K m, 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 distributi on 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 geomet ry factor, J, using Equation 13. The only remaining variable needed is Yk, the tooth form f actor, 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],
....= 23 .......... .... 1 .... tan.... 3.... Equation 35 where kf is the actual stress concentration factor, derived from the theoretical stress concentration factor, kt, Fn is the angle which the normal force makes with a li ne perpendicular to the tooth centerline, Xn is a ratio which defines the gear toot h strength factor, and tn is one-half the tooth thickness at the critical section of the ge ar 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 . less than .5 as a recommended initial value. In this applicati on however, because of the limitations of the formulas above, a value for the ratio does not
turn positive until . 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 . leads to a range for . between .05 and .06. Table 9 below displays computed results using assumed value s for . in the range of .05 to .06, and the effect this has on the important factors i n the bending stress analysis. Table 9 - Assumed values for . and its effect on bending stress . 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 . was assumed and the remaining values for Xn, Yk, J, and Bending Stress were calculated based on the assumed . values. Because the ultimate goal in this analytical procedure is not to reduce the bending stress b ut to accurately calculate the bending stress, a . value of .05935 will be used becaus e 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 bendin g stress and is therefore considered minimal. Therefore, a J value of .24205 will be subs tituted into Equation 8 previously discussed, in order to calculate the bending stress i n the gear teeth. This results in a value for fb of, ....=31.5 ...... 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 compi led 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
Teeth. This extracted data represents the results of laboratory and field experi ence for the specified material and condition of that material, and is greatly dependent on material composition, cleanliness, quality, heat treatment, mechanical propertie s, forging practices, residual stress, final processing operations in manufacture and metho d of stress calculation [8]. Discussion of these factors is covered in the AGMA stand ard and should be adhered to in order for these values to accurately represent the desig n and analysis covered throughout this paper. Because the design and analysis presente d 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 A GMA standard which provides hardness recommendations for the core of the gear as wel l 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 h ardness 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 comp ared. As such, an allowable value for compressive stress is 250 ksi which is much high er 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 ks i. This
analysis proves that the finalized gear tooth design has been shown to be safe f or operation in this specific application. 3.5 Gear Life Calculations Gear life calculations were performed in accord with AGMA recommendations, specifically utilizing the Miners rule methodology presented earlier in Section 2 .6. The flight spectrum, presented in Appendix A, shows five flight maneuvers in which f atigue damage occurs to the gear teeth. Using a Microsoft Excel spreadsheet specificall y designed for the calculation of damage accumulation, the five maneuvers were inp ut into Appendix D, along with the composite percent time and input power associated wit h each of the five maneuvers. The input power is then converted to a bending stres s 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 d amage 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-T HENELSE 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 hour s. If the calculated fatigue life is greater than 50,000 hours, the gear is said to have u nlimited life for application in this helicopter. As Appendix E shows, the gear designed and a nalyzed herein has unlimited life for both bending life and durability life.
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 intermed iate gearbox of a medium class helicopter and was framed around the existing transmis sion components in use, specifically utilizing the current transmission housings, bea rings, 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 pro per selection of certain design parameters. Upon completion of the design phase of t he gear, analysis was conducted to ensure appropriate margins of safety had been implemen ted 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 criti cal sections of the gear. Margins of safety were calculated at the two critical sect ions 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 Figu re 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 enduranc e 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 sa fety equal to .87. A positive margin of safety was shown to provide adequate safety f or 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 wer e calculated using iterative procedures that were explained in detail. Hertz (comp ressive) and bending stresses in the gear teeth were calculated using the recommended pra ctices 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 k si. Per the AGMA standards, allowable stresses in carburized and case hardened gear teet h are
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 lo ad spectrum and the specific flight maneuvers that cause fatigue damage to the gear teeth. Miners 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.
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. Machinerys 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 Beve l, 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.
6. Appendix A Helicopter flight spectrum with anticipated horsepower and torque loads. See Mic rosoft Excel file titled, Flight Spectrum and Anticipated Load Conditions on the associat ed CD.
7. Appendix B Fatigue and static analysis on gear shaft. See Microsoft Excel file titled, Fatig ue and Static Analysis on the associated CD.
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.
9. Appendix D Fatigue life calculations. See the Microsoft Excel file titled, Spiral Bevel Gear Life Calculation on the associated CD.
