Aero Engine Accessory Gearbox - MECH 393, McGill Universityy

McGill University MECH 393: Machine Element Design

Aero Engine Accessory Gearbox

FOR

Professor Pascal Hubert

BY Imran Jameel Julien Brillon John Cram

Executive Summary

This report is a presentation of the preliminary design of a compact aero engine accessory gearbox. Our gearbox receives power from a radial shaft connected to the main aero engine turbine shaft. It is designed to be compact, lightweight and easy to assemble as speci ed in the mission statement provided to us by the client.

This report includes a detailed design of the input shaft and idler shaft for our gearbox.

This report will also detail the speci cations for all gears and bearings selected for our design. This report also includes a preliminary design of a housing for the gearbox design complete with the required fasteners to mount the gearbox to the chassis. Images of the proposed design are attached to provide a visual representation of the gearbox and all of its components shown in both assembled and exploded views. Complete dimensions of these components are included in this report via manufacturer drawings. Furthermore, a summary of all of the parts required for this design can be found in the Bill of Materials.

The report begins with a review of the design speci cations provided in the mission statement by the client. Following the design speci cations is a theoretical explanation of the method used to calculate the gear reduction ratios applied by the gearbox. Using this information a detailed explanation of the gear, shaft, bearing, casing, and fastener analyses is provided. Justi cation of all assumptions and reasoning behind said analyses is included in addition to a thorough calculation of safety factors to ensure failure is avoided. Finally, a simple assembly and disassembly procedure is reviewed before some concluding remarks regarding the overall design process and possible improvements are mentioned.

1

Table of Contents Executive Summary

1

List of Figures

4

List of Tables

4

1. Introduction

5

1.1 Problem Statement

5

2. Theoretical Development

7

2.1 Determination of Gearbox Ratio 2.2 Evaluating Gearbox Cycle 2.3 Determination of Fatigue and Static Loading Case 2.4 Initial Gearbox Layout

7 9 9 10

3. Results

11

3.1 Gear Analysis 3.1.1 Gear Train Layout and General FBD 3.1.2 Material Selection, Surface Treatment, and AGMA Stress Numbers 3.1.3 AGMA Factors and Stresses 3.1.4 Load Cycles 3.1.5 Gear Parameters 3.1.5.1 Bevel Gear 3.1.5.2 Spur Gears 3.2 Shaft Analysis 3.2.1 Initial Layout and Design 3.2.2 Material Selection 3.2.3 Critical Location Analysis 3.2.4 Key Analysis 3.2.5 Shaft De ection Analysis 3.2.6 Critical Speed Analysis

3.3 Bearing Analysis 3.3.1 Bearing Load Analysis 3.3.2 Bearing Load Ratings and Equivalent forces 3.3.3 Bearing Selection 3.4 Casing and Fasteners Analysis 3.4.1 General Casing Layout 3.4.2 Material Selection 3.4.3 Fastener Layout 3.4.4 Bolt Selection 3.4.5 Fastener Failure Analysis 3.5 Gearbox Oil 3.6 Overall Weight Estimation

11 11 12 12 15 17 18

19 19 20 20 22 23 25 25 25 26 28 28 28 28 28 29 29 30 30

4. Assembly and Disassembly Procedure

31

5. Conclusion

32

2

Appendix A: Shear, Bending Moment and Torque Diagrams

35

Appendix B: Catalogues

36

B1. Shaft key standard imperial dimensions B2. Typical Maximum Ranges for Slopes and Transverse De ections B3. Weibull Parameters Rating Lives: B4. Radial Load Factors for Ball Bearings B5. SKF Bearing Catalogue

36 36 37

Appendix C: 3D CAD

42

Appendix D: Machine Drawings

44

37 38

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List of Figures Figure 1: Accessory Gearbox Layout Figure 2: Accessory Gearbox on a Turbine Figure 3: Initial Gearbox Layout Figure 4: FBD of Gear and Transmitted Forces Figure 5: Component Alignment on Shafts Figure 6: Eccentric Loading Diagram

7 7 11 13 21 30

List of Tables Table 1: Mission Speci cation Table 2: Power and Speed Speci cation Table 3: Dynamic Phase Loading Table 4: Gear Pair Analyses Table 5: Number of Cycles for Gears and Bearings Table 6: AGMA Stress Summary Table 7: Safety Factors for Bending and Contact Stress Table 8: Bending and Contact Strengths for Grade 3 Carburized and Hardened Steel Table 9: Summary of Load Cycle Factors Table 10: Bevel Gear Speci cations Table 11: Gear Selection Options Table 12: Final Gear Speci cations Table 13: Minimum Number of Teeth on Pinion Table 14: Critical Location Analysis Table 15: Estimated Stress Concentrations Table 16: Key Analysis Table 17: Key and Keyway Design Table 18: De ections and Slopes at Critical Locations Table 19: Critical Speed Analysis Table 20: Bearing Load Analysis Table 21: Bearing Load Rating Analysis Table 22: Equivalent Forces on Input Shaft for Bearing Analysis Table 23: Bearing Selection Table 24: Fastener Analysis Table 25: Estimation of Total Gearbox Weight

7 8 9 10 11 15 17 17 18 18 19 19 20 22 23 24 24 25 26 27 28 28 29 31 32

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1.

Introduction

Our design process began with an analysis of the mission speci cation of the four phases (Idle, Takeo , Cruise, Landing) which the accessories (Starter/Generator, Gearbox Input Shaft, Hydraulic Pump, Oil Pump) and input (Engine Radial Shaft) would need to endure with respect to the performance data of the aero engine.

The drive cycle was re ned to establish the input torques at each phase based on the maximum speed and power of the accessories and input. At this point we set up a generalized layout of our gearbox components and computed the loads generated in all shafts, gears and bearings based on shaft lengths that could accommodate the components of the shaft. These loads were used to select our gear dimensions using AGMA gear calculations. Safety factors for all gears were calculated to ensure no failures. We then selected suitable shaft dimensions based on the Modi ed Goodman failure criterion for fatigue failure and a modi ed, but more conservative, version of DE failure criterion for static failure. Following this, we selected appropriate keys, collars, retaining rings and bearings from standard manufacturer catalogs for our shafts. Lastly, we established the design of our housing and the required fasteners to proper mounting to the engine.

In this report we have included details about the gearbox’s requirements and the design decisions made during this process. We’ve also included details of the computations performed to verify the safety of our design and CAD models and drawings of all components present in our nal gearbox. We conclude with a summary of potential issues with the current design and avenues to potentially improve its performance.

1.1.

Problem Statement

A series of design requirements for the aero engine accessory gearbox were provided to our team.

The gearbox is to be compact, lightweight, and easy to assemble. The gearbox must drive the following accessories: a starter/generator, a hydraulic pump, and an oil pump. The gearbox gets power from a radial shaft connected to the engine input shaft by a set of bevel gears. We were not required to design the input bevel gear. This is demonstrated schematically in the gearbox layout provided in Figure 1.

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Figure 1 - Accessory Gearbox Layout

Figure 2 - Accessory Gearbox on a Turbine

We are also provided with the mission speci cations.

Table 1 - Mission Speci cation

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Furthermore, we are given the the maximum speeds and power for the accessories and input.

Table 2 - Power and Speed Speci cations

Additional design factors provided by the client include:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

2.

The accumulation of tolerances from the manufacturing methods accounts for an overload of 10%.

The available tooling for the gears is for diametral pitch of 10, 12 and 16; and angle of pressure of 20°.

Assume a diametral pitch of 16 and diameter of 3” for the input bevel gear.

Life for the gears is 30,000 hour and 10,000 hours for the bearings.

Safety factor for shafts is 1.5 Gears are made of carburized steel, hardened to 55 min HRC, AGMA quality class 8.

Misalignment: .0005 in/in.

The reliability is assumed at 1 failure per 1000 ights.

The lubrication system operation temperature is 245°F. The oil type used is MIL-23699 type II. The oil temperature is 210°F at the entry and 280°F at the exit.

Assume an engine outside diameter of 60”.

Assume that the accessories have a diameter of 4” and are attached to the gearbox casing.

Theoretical Development

In this section, we will explain the steps taken by our team to design the accessory gearbox while meeting all of our clients needs.

2.1.

Determination of Gearbox Ratio

To begin the design process, we rst needed decide which gear combination was optimal. We were given the maximum input speeds and power for each shaft, as well as the percentage of maximum engine power. From this we determined the “Input torque” on the Input, Idler, A, and B shafts where the static load case is represented as “Engine Start”. The torque transmitted to the other shafts during engine start is negligible compared to the input torque and are thus taken to be non-existent:

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Table 3 - Dynamic Phase Loading

Considering our initial gearbox layout we performed the following gear ratio calculations:

N1 N2

=

T orque on Input Shaf t T orque on Shaf t A

=

25000 lb f t 12500 lb f t

=2

Since the torques on the input shaft and shaft B are all proportional to the % of maximum engine power in each phase, the gear ratio is constant. We are not given any information on the idler shaft, thus we must analyze these shafts (input, idler, and shaft A) together as a compound gear train.

e=

nl nf

=

900rpm 8000rpm

= NN 31 NN 54

Thus,

N 5 = N 1 NN 34 8000 900 (1)

Note: speed values from the cruise phase are used in this case since for the hydraulic pump, 20% of maximum engine power is used and yields the highest allowable compound gear ratio.

Knowing that: 10" ≥ Distance between shaf t A and Input ≥ 5"

10" ≥ 0.5(d1 + d3) + 0.5(d4 + d5) ≥ 5" (2)

N = P d (3)

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To minimize gear weight and overall size we tried to use appropriate standardized gear diameter values keeping equation (2) as close to the lower limit as possible while satisfying both equations (1) and (2) using the relation (3). We found that using a ratio of N3/N4=8/3 gave integer values for N5, which is desirable for gear teeth.

Thus, we arrived at the following results:

Table 4 - Gear Pair Analysis Note: distance and diameter values are in inches.

2.2.

Evaluating Gearbox Cycle

2.3.

Determination of Fatigue and Static Loading Case

The cycle of our gearbox is taken as the cycles for one ight. The engine and its attached accessory gearbox must operate for the duration of the startup, take o , idle, cruise, and landing phases of the ight or cycle. This information is stated by the client in Table 1 above.

Thus, one cycle of the gearbox is obtained by summing the durations of each phase of ight to a duration of 210 minutes or 3.5 hours.

The torque transferred to shafts and devices other than the input shaft and starter/generator during the Engine Start phase is assumed to be negligible due to the small time span of this phase.

Accordingly, this phase is taken as the static loading case with an input torque of 350 lbf-in on the input shaft.

In terms of fatigue, the gearbox is designed for a nite life of approximately 2850 cycles at the loads given for each phase. This information was determined using the given life (in hours) of the gears and the bearings from the client and is brie y summarized below.

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Table 5 - Number of Cycles for Gears and Bearings

2.4.

Initial Gearbox Layout

The solution to the problem speci ed by the client can be approached in a number for ways. For example, in terms of the gearbox layout the idler shaft might hold one gear or it might hold two gears or it may not be present in the nal solution whatsoever. We brie y explored the feasibility of a number of these solutions and settled on the following initial layout. This layout allows for a minimal distance between shafts.

Figure 3 - Initial Gearbox Layout

Lengths of shafts B and A have been minimized to reduce the overall gearbox weight contributed by the shafts themselves in addition to the volume of material that can be excluded from the casing as a result of not needing to seal in those extra shaft lengths.

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3.

Results 3.1.

Gear Analysis

The information calculated in the Determination of Gearbox Ratio section above presents us with a number of options for our various gear pairs. Note that Pair 1 denotes the link between the input and B shaft, Pair 2 denotes the link between the input and idler shafts, and Pair 3 denotes the link between the idler and A shafts. All potential gear pairs were analyzed and compared rstly on the basis of load and safety factors. It was found that no particular gear or gear pair exceeded any allowable loads and or safety factors. A secondary comparison particularly on basis of the speci ed design criterion of a compact and lightweight gearbox allowed for a selection of appropriate gears.

The following gear results section will provide a complete review of the parameters and loads characterized by these gears. These gears are of carburized steel, hardened to 55 min HRC, and AGMA quality class 8 as speci ed by in the mission statement.

3.1.1.

Gear Train Layout and General FBD

A gear train layout was not speci ed in the mission statement so there were several possibilities for how to transmit the input torque to the required devices. Keeping in mind the need for a compact gearbox we opted for the general layout seen in Initial Gearbox Layout that includes one gear on all shafts excluding the idler shaft, which has two gears. As the reduction between the input shaft and shaft A is quite large, the receiving gears would have to be of a signi cant diameter which would increase the distance between our shafts. Two gears on the idler shaft allows for a distance closer to the lower limit of 5”. It was noted that the additional gear G4 will add extra weight, however it will be of small diameter and will also allow for G5 to be of smaller diameter than if the idler shaft only had one larger gear. Overall, the di erence in weight between the two options should be relatively small.

A general free body diagram of our gears and their transmitted forces using a pressure angle of 20° is shown below. The shafts are in order of B, input, idler, and A from top to bottom of this diagram with the weights of the gears acting downwards.

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Figure 4 -FBD of Gear and Transmitted Forces

3.1.2.

Material Selection, Surface Treatment, and AGMA Stress Numbers

The mission statement speci es gears are of carburized steel, hardened to 55 min HRC, and AGMA quality class 8. These These characteristics ensure high overload capacity, quality, and precision. The calculation of all related factors and loads have taken these characteristics into account.

In order to calculate the approximate weight of our gears manufactured using this material, we needed to nd the density of said material. A density of 7.98 g/cm3, or 0.288 lbm/in3, was estimated based on the physical properties of a high quality carburizing steel used for aerospace power transmission applications1.

3.1.3.

AGMA Factors and Stresses

This section will outline the assumptions and reasoning behind the selection of the various AGMA factors, concluding nally with the corresponding stresses for each gear. For constants requiring no assumptions or selections and instead being a simple calculation using known values, refer to the attached spreadsheet. Detailed numbers and calculations for all gears can be found in the “Gears” sheet under “AGMA Method”.

Note that the nomenclature for our gears follows a system of three gear pairs each consisting of a pinion and gear. For example, the gear on shaft B (G2 on the “Initial Gearbox Layout” is part of Pair 1 and is the Gear, so is labeled as “1G”).

1

http://www.questek.com/filebase/src/Material_Data_Sheets/FerriumC64CarpenterDataShe.pdf

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In order to calculate the bending and contact stresses of our gears we used the following AGMA standard relations:

Transmitted forces WC were derived from a pressure angle of 20° using the applied loads on the shafts.

Overload factor Ko was selected as 1.5 based o of a moderate shock driven machine and a light shock power source. These values should be appropriate and perhaps slightly conservative considering the impact on an airplane from the generated thrust, the landing impacts, and turbulence.

Dynamic factor KV is calculated using the pitchline velocities of the gears as well as with constants A and B determined by the AGMA quality number of our gears.

Size factor KS can usually be assumed as 1 for standard gear sizes, but for precision we calculated these individually using the appropriate face widths, lewis factors, and pitch for each gear.

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Load distribution KM was calculated using a number of constants that can be found under “KM Constants”;Crowned and uncrowned gears were assessed, straddle mounted pinion with S1/S ratio greater than 0.175 were used according to our initial gearbox layout. It was assumed that our gears were not adjusted at assembly, nor was compatibility improved.

Empirical constants A, B, and C were chosen for precision enclosed units as this is what we have in our accessory gearbox according to the quality number of 8 indicating precision gearing.

Rim thickness factor KB is taken as 1.0 for a rim thickness below the tooth to tooth height ratio of greater than or equal to 1.2. At the time of gear selection these dimensions were not accurately known, however by taking into consideration the outside diameters of our gears relative to approximate bore diameter we can take this as a reasonable assumption.

Geometry factor J depends on face-contact ratio mN which is taken as zero for spur gears.

Consequently J values are taken from the following table using appropriate number of teeth:

Surface condition factor Cf is taken as unity due to no known detrimental surface nish e ects.

Table 6 - AGMA Stress Summary

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3.1.4.

Load Cycles

The mission statement provided us with information regarding the load cycles of the gears. Life for the gears is 30,000 hours. Using this lifetime coupled with the cycle time of one ight, or 3.50 hours, we nd that the gears must complete a total of 8572 cycles in their lifetime.

As typical AGMA strength and stress factors are values are based on 107 number of cycles, we calculated appropriate YN and ZN values using Tables 14-14 and 14-15 below to modify them based on the number of cycles our gears must endure. The “Case Carb.” data was used for our gears as they are made of carburized steel.

Reliability factor KR was chosen corresponding to a reliability of 1 failure in 1000 ights, or a reliability of 0.999. Temperature factor KT was assumed for a temperature below 120 ° celsius. This assumption is accurate considering the average operating conditions of an airplane while also taking into account heat generated within the engine and the gearbox itself from combustion, friction, etc.

Rim thickness factor KB is taken as 1.0 for a rim thickness below the tooth to tooth height ratio of greater than or equal to 1.2. At the time of gear selection these dimensions were not accurately known, however by taking into consideration the outside diameters of our gears relative to approximate bore diameter we can take this as a reasonable assumption. These factors are used to

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calculate the safety factors SF for bending stress and SH for contact stress of our gears using the following relations:

Table 7 - Safety factors for Bending and Contact Stress

As SH3 >> SF we can conclude that the biggest threat for failure is in bending. However, all safety factors are acceptable so there is no issue with the selected gears.

Note that in our analysis of crowned and uncrowned gear teeth we found uncrowned stresses to be consistently greater than crowned stresses for every possible gear. Hence the stresses used in the relations above are those of crowned teeth. These numbers can be seen in more detail under the “Gears” sheet of the attached spreadsheet.

Repeatedly applied bending and contacts strengths ST and SC were chosen for grade 3 carburized and hardened steel:

Table 8 - Bending and Contact Strengths for Grade 3 Carburized and Hardened Steel

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A summary of the load cycle factors is given in the table below:

Table 9 - Summary of Load Cycle factors

3.1.5.

Gear Parameters

3.1.5.1.

Bevel Gear

Gear “B”, the bevel gear providing the input torque from the radial shaft of the engine, was not to be designed by us as speci ed in the mission statement. In addition, no contact and bending stresses are calculated for this gear due to a lack of information provided in the mission statement by the client, as this gear is not to be designed. However, its weight was indeed needed and was calculated as though it were a spur gear with a face width of 0.75 in (standard for gears of diametral pitch 16):

The forces received on this pinion from its mated radial shaft bevel gear was calculated using standard equations from bevel gear analysis. The rpm reduction was given, in addition to the diameter and diametral pitch, therefore the number of teeth could be determined. Assuming a pressure angle of 20 ° the axial and transmitted forces were found and summarized below: Table 10 - Bevel Gear Speci cations

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It is to be noted that a keyway design and analysis is not provided for this input bevel gear as it will be press tted onto the input shaft at a point outside of the gearbox casing.

3.1.5.2.

Spur Gears

Following a complete weight, size, and stress analysis of all possible gear combinations, we nd that Option 3 and Option 6 give us the best bang for our buck. Minimal distances between shafts means a smaller casing pro le and minimal total gear mass means less overall weight. The crux of Option 3 however is the tradeo for a very marginal di erence in distance and weight comes at the cost of increases stresses on gears. Due to the relative equality of the size and weight of each option we chose to proceed with Option 6.

Note that the “Total Mass” column in the table below is the total mass of all gears excluding the bevel gear, as its mass remains constant over all gear options.

Table 11 - Gear Selection Options After selecting Option 6 we were able to pinpoint the nal speci cations for our gear train:

Table 12 - Final Gear Speci cation

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Lastly, as a nal precaution the minimum number of teeth on the pinion of each gear pair is checked so that these gears do not have any interference between each other in terms of gear meshing. The follow relation was used, where represents the pressure angle of our gears:

Table 13 - Minimum Number of Teeth on Pinion

These values are much less than the actual number of teeth on our pinions, thus there is no interference and these gears will be used for all subsequent design considerations.

It is important to note that the speci ed overload factor of 10% is not explicitly applied to these loads and therefore not accounted for in this analysis, however it is indeed accounted for via a number of conservative estimations and nal selections regarding shaft dimensions that are signi cantly greater than the critical values.

3.2.

Shaft Analysis

This section outlines the results of the procedure that was used to determine the dimensions, features, and overall design of all of the shafts included in the gearbox assembly and reports the .

The majority of these calculations and reasoning was done by hand on paper and therefore cannot be referenced on the attached spreadsheet, but their ndings are well documented in the tables that will follow.

3.2.1.

Initial Layout and Design

Before any dimensions could accurately be decided, it was necessary to assume some initial layout of the shafts from which to base our calculations o . Using the “Initial Gearbox Layout” in Section 2.4, we were able to determine what features our shafts would require to axially locate its components (ie. shoulders, retaining rings, grooves, keyways, etc.) as well as some estimated distances between them in order to avoid interference. In order to retain some consistency in these initial assumptions between shafts we chose conservative bearing widths of 1.0 in, groove widths of 0.25 in, and thus established lengths from bearing center to gear centers that will line our gears up with each other as shown below:

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Figure 5 - Component alignment of shafts

As the assumed lengths of standard components change, the lengths of other parts of the shaft will change such that these distances remain the same; this way we will not need to recalculate loads.

Arbitrary extra lengths of shaft B and shaft A are included in the drawing, but aside from leaving space for a retaining ring these lengths will not be present in our nal design. These shafts are shorter than the input and idler by design; it should allow for a gearbox of reduced size.

3.2.2.

Material Selection

The material the shafts are to be machined with plays an integral role in calculating the necessary dimensions of the shafts to avoid both fatigue and static failure. As an initial material to begin our calculations from we chose an AISI 1020 CD Steel that is relatively inexpensive and common for shaft applications. We were satis ed with the results this material gave and therefore there was no need to increase yield or ultimate strengths by selecting a stronger steel.

3.2.3.

Critical Location Analysis

The approximate shaft lengths in the previous section allow for calculation of shear, moment, and torque diagrams from which we can determine critical diameters at locations of high stress. The

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resulting moments, torques, diameters, and static safety factors at suspected critical locations were are summarized below. An initial ASME diameter estimation was calculated using Cm and Ct values of 1.75 and 1.25 respectively, which are the middle range values for minor impacts used considering the application of the gearbox. This approximate d allows for an approximate kB calculation.A reiteration using this diameter was done using the Modi ed-Goodman Criterion in addition to the speci ed shaft safety factor of n = 1.5 from the mission statement to nd the critical MGM diameter at that location. In addition, this new MGM diameter is used to reiterate upon the initial kB approximation which were all found to be within 0.05 of the approximated values and thus justifying the decision to not reiterate further.

Table 14 - Critical Location Analysis

Note: The static check safety factors were calculated using the conservative maximum von mises stress (ie the sum of alternating and midrange von mises stresses).

The values in the above table are based o of the following estimated stress concentrations: Table 15 - Estimated Stress Concentrations

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By taking the largest critical diameter of each shaft and increasing it to the closest standard diameter in english units we were able to determine the diameters for all points along the shaft. A sample of this process for the input shaft can be seen in the “Shafts” sheet of the attached spreadsheet. Since the diameters are not decreasing, we are still operating above the speci ed safety factor. Note that these values are not explicitly tabulated at this point as they are still subject to change after analyzing the de ection analysis and accommodating the bore diameters of selected bearings. These nal dimensions can be seen in the attached part drawings, along with the llet radii at shoulders which are consistent with the estimated Kt and Kts used to nd our diameters.

It is important to note that the weight of the shafts themselves have not been taken into account during this procedure. The reasoning behind this decision is as follows: These weights cannot be accurately known until after the critical location analysis due to their dependence on the nal diameters chosen. Knowing this, we can forego taking into account their weights by making conservative choices later in the design stage (bearing selection, for example) to accommodate a slightly increased load. These added weights can be included in the nal estimated weight of the gearbox as a whole. If we model the shaft as some average nominal diameter of 1 in and of length 4.5 in we can can calculate the total weight of this shaft to be about 1 lbm. To account for 4 shafts and some degree of error we can con dently take the total weight of our shafts to be about 5 lbm.

3.2.4.

Key Analysis

Based on the dimensions of each shaft, an appropriate key was chosen (using the table in Appendix B1). Speci cally a square key was chosen as square keys are generally used for small shafts. 1020 CD steel with a yield strength of 57 kpsi was subsequently chosen for the key material. We then assigned key lengths (L_max) such that they match the face width of the gear. However since the keyways will need to be machined into the shaft, we assumed that the rounded edges of the keyway would not carry a signi cant load. Therefore, we did all our computation for keys using a corrected length (L_corrected) equal to the total length of the key minus the width of the key, in order to nd the at portion of the key in contact with the gear. We then veri ed its safety factor for crushing by bearing failure using the equation below. Since a square key was chosen, checking only the crushing failure was adequate since the shearing failure will be less critical according to the distortion energy failure theory, and equal according to the maximum shear stress failure theory. The table below summarizes our ndings.

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Table 16 - Key Analysis

Table 17 - Key and Keyway Design

Splines were initially proposed to transmit the torque between our shafts and gears, however due to increased complexity and no signi cant increase in the e ectiveness of our design this proposal did not make the cut for our nal design, just as the ‘Habs’ did not make the cut for the 2016 Stanley Cup Playo s.

3.2.5.

Shaft De ection Analysis

Assuming a uniform cross section, beam de ection equations and the principle of superposition (POS) were applied to analyze the de ections at various locations along the shaft. Shaft A and B were modelled as simple supported beams with an intermediate load, whilst the Idler Shaft was modelled as a simple supported beams with two intermediate loads by using the POS. Lastly, the Input Shaft was modelled as a simply supported beam with an overhanging load caused by the weight of the bevel gear, from the centre of the outermost bearing to that of the bevel gear. The gures and formulas below illustrate the approach we took.

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The table below illustrates the de ections and rotations at the critical components on the shafts, such as the gears and the bearings.

Table 18 - De ections and Slopes at Critical Components

Using Table 7-2 in Appendix B2, the bearings slope and gear de ections were compared to typical maximum ranges for slopes and transverse de ections. From this analysis, we concluded that all our shafts were within the allowable ranges for spur gears and deep groove ball bearings, except for the Idler Shaft. Since the second bearing on the Idler Shaft was out of range, we concluded that a larger diameter shaft was warranted. As a result, calculations for a new diameter (dnew) were made and the ratio of the new and old diameter(dold) was found. We then multiplied all diameters by this ratio to nd our new dimensions.

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3.2.6.

Critical Speed Analysis

At certain speeds, the shaft is unstable, with de ections increasing without upper bound. This speed known as the critical speed must be analysed for all shafts in operation. We seeked critical speed at least twice the maximum operating speed as a value to be considered safe. This critical speed ( ω ) was calculated using the following formula.

Table 19 - Critical Speed Analysis

3.3.

Bearing Analysis

The shaft layout is designed such that a shoulder will be pressed against both bearings. This will help with the assembly of the gearbox as the shafts will perfectly slide into place. After considering our shaft layout, we selected deep groove ball bearings throughout the design in order to withstand both the radial and axial loads.

Reviewing the client’s design speci cation for an overall reliability (R) of 99.99%, we rearrange the following equation to solve for the individual reliabilities of each bearing. Note A and B refer to the left bearing and right bearings respectively.

Thus, RA = RB = √R = √0.999

3.3.1.

Bearing Load Analysis

The bearings are subject to variable loading during the various driving phases of the engine.

However, we computed the maximum radial loads (on all shafts) and axial loads (for the input shaft) using FBDs and the table below summarises the radial and axial loads on the shaft at that instant.

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Table 20 - Bearing Load Analysis

3.3.2.

Bearing Load Ratings and Equivalent Forces

In this section of the bearing analysis, we will nd the minimum catalogue rating (C10) value needed for our selected bearings from the SKF catalogues. We used the following formula to determine the load rating of a bearing:

where af is the application factor, a is the constant previously determined to be 3 since we will be using ball bearings for our gearbox, FD is the applied force on the bearing, RD is the reliability of the bearing which in our case is √0.999 . x0 ,θ and b are known as Weibull Parameters from

Manufacturer 2 (in Appendix B3). xD is the multiple of rating life which is de ned as where LD =60 L d n d is the design life of the bearing in revolutions and LR = 1(106) is the rating life of the bearing.

The client requires a Ld = 10,000 hours.

In our design, the bearings at the shaft B must survive a life of 3x109 revolutions and the bearings at shaft A have a design life of 2.7x109 revolutions.

Finally, with the radial forces calculated in the load analysis of the bearing analysis, the minimum catalogue load rating, reliability application factor, and multiple of rating life for each bearing are:

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Table 21 - Bearing Load Rating Analysis

Load application factors are for “machinery with light impact”. This is an accurate assumption due to the typical impacts airplanes receive during landing and periods of turbulence. The values presented above for C10 represent the minimum catalogue values that would need to be used based solely on the radial load experienced by the bearing. However, during operation, the input shaft will need to support an axial load due to the bevel gear. We thus need to consider what the required C10 value of each bearing on the input shaft would be if it needed to carry an axial load. Doing so requires computing an equivalent force using the following formula:

In our case since the inner ring of the bearing rotates V = 1. Xi and Yi terms are constants, which can be found in the table in Appendix B4. As shown in the table, the value of the constants depend on the magnitude of e. Therefore, to compute the equivalent force on the bearings, an iterative bearing selection process must be done to obtain the appropriate Co value. Once the bearing is selected, the calculated e and Fa/Co values will determine the correct X and Y values. Next, interpolation was used to compute any values that may be between the tabulated values. Finally, using the same formula presented above we computed the C10 value. The reliability of the bearing was then recalculated to ensure that it surpassed our design requirement (combined reliability of 0.99956).

Table 22 - Equivalent Forces on Input Shaft for Bearing Analysis

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3.3.3.

Bearing Selection

The bearing’s bore must be dimensioned such that it conforms with the shaft dimensions selected previously and that they withstand the loads applied by the gears. Thus, based on our bore diameter (d) and our dynamic (C) and static load rating (C0) calculations we chose the following bearings from the SKF catalogue (in Appendix B3).

Table 23 - Bearing Selection

3.4.

Casing Analysis

This section outlines the design of the casing, the points of attachment, and the selection of fasteners used to hold the gearbox together in addition to the fasteners used to mount the gearbox to the engine.

3.4.1.

General Casing Layout

It is speci ed by the client that the inner diameter of the engine to which the gearbox is to be mounted is a diameter of 60’. The casing has been designed with a curved inner surface to allow for it to mount ush to the engine. Pockets have been included in the thickness of the case to support bearings that will be press t within. As this is not possible for the shafts of smaller lengths (A and B), there are custom supports to hold the bearings on these shafts that would otherwise be oating. The complex geometries of this case can be achieved with a standard casting process.

3.4.2.

Material Selection

A standard lightweight high strength aluminum has been selected for the casing to minimize the weight contribution from the case while achieving high strength and durability. Aluminum alloy 7075-T6 with a yield strength and an ultimate strength of 73 ksi and 83 ksi respectively. This material also has decent machinability which aids in the manufacturing process of the case. It has a relatively high cost, but as cost is not a limiting design factor imposed by the client this does not pose an issue.

The density of this material is approximately 0.102 lb/in2.2

3.4.3. Fastener Layout

- Blurb explaining use of less fasteners to hold the casing halves together in order to reduce overall assembly time -

2

http://asm.matweb.com/search/SpecificMaterial.asp?bassnum=MA7075T6

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3.4.4.

Bolt Selection Assuming a turbine thickness of ½’’, the selected bolts to fasten the gearbox to the outside of the 60” turbine consist of Extreme-Strength Grade 9 Steel Cap Screw 3/4"-10 Thread, 3" Long, Zinc-Plated and Cadmium Yellow-Chromate Plated Steel Hex Nut - Grade 9, 3/4"-10 Thread Size, 1-1/8" Wide, 25/32" High from McMaster Carr.

Considering a casing thickness of 1.5”, the protruding length of the screw without a washer is 0.21875” (3”-1.5”-0.5”-25/32”=0.21875”). Four bolts have have been used to symmetrically mount the gearbox to the aero engine. In the case that the turbine thickness is greater or less than that assumed, the client has the option of screw lengths ranging from 1.5” to 10” while keeping all other bolt speci cations the same.

3.4.5.

Fastener Failure Analysis Assuming the gearbox is xed in the position of maximum shear on the bolts (i.e. such that the bolts are approximately parallel with respect to the ground), the following eccentric loading analysis has been performed:

- Weight of Gearbox=3952.3lbf - Using the appropriate material densities in SolidWorks, the center of mass is given as (x,y,z)=(0.81, 0.00, -2.30).

- Minimum yield strength of bolts = 180kpsi (McMaster Carr) - Minimum yield strength of casing = 73kpsi - Diameter of bolts: 0.75in - Thickness of casing: 0.5in

Figure 6 - Eccentric Loading Diagram

Since the di erence of the distance between bolts and centroid for each bolt is small, it has been approximated as the mean distance to simplify force analysis:

Ra=Rb=<R>=(Ra’+Rb’)/2=12.782401 in

By symmetry, F’a=F’b=F’

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F”a=F”b=F”

And by summing static forces and moments, ΣFy=0; F’=V/2=1976.15 lbf ΣMy=0; F”=M/2r=606.03 lbf Since two bolts carry the force at A and B, the resultant maximum shearing force on the bolts is:

Fmax=√{(F’/2)2+(F”/2)2} = 1033.5 lbf

Using the following relations: As= d2/4 Ab=t*d Ssy=0.577*Sy = Fmax/As = Ssy/n b = Fmax/Ab = Sy/n

The factors of safety calculated for the following failure modes using the data and relations above have been summarized in the table below:

Failure mode

Factor of Safety

Shear on bolt

44.4

Bearing on bolt

196

Bearing on casing

79.5

Table 24 - Fastener Analysis Results

Since all factors of safety are greater than unity, these fasteners are acceptable.

3.5.

Gearbox Oil

As speci ed in the mission statement, the selected lubricant is MIL-23699 type II for operation at 245°F. The oil temperature is 210°F at the entry and 280°F at the exit. The relatively high operation temperature is due to the heat generated inside of the engine.

3.6.

Overall Weight Estimation

The total weight of the complete gearbox assembly including gears, shafts, casing, and fasteners is estimated below, where the weight of the casing and its fasteners was calculated using Solidworks:

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Table 25 - Estimation of Total Gearbox Weight

4.

Assembly and Disassembly Procedure

The gearbox assembly revolves around the design of the casing, which consists of two asymmetrical halves. The input, idler, and shaft A assembly procedures are straightforward and rapid: simply slide all components (retaining rings, gears and keys, etc.) onto the shafts from their respective sides (ie do not slide a gear on from an entry position where a shoulder is located between the gears intended destination and its starting point). This can be achieved by working from smaller diameters up to larger diameters as is intended by the shafts design. Before the casing can be enclose these shafts however, a slightly more involved procedure for shafts and B must be carried out. After preparing the shafts, the two halves of the casing can be pressed together and fastened using the selected fasteners (20). Next the input bevel gear and its collar and retaining ring are installed before the whole gearbox is mounted onto the aero engine using the selected fasteners (4). A complete step by step procedure is given below:

1.

Prepare shafts for enclosure a. Input Shaft i.

Slide all components onto shaft excluding components located outside of gearbox (bevel gear, collar, etc) ensuring entry from the correct end of shaft b. Idler Shaft i.

Slide all components onto shaft ensuring entry from the correct end of shaft c. Shaft A i.

Slide all components onto shaft ensuring entry from the correct end of shaft d. Shaft B i.

Line up in the following order the centers of the gear, its self locking retaining ring, the left bearing and its retaining ring (bearing and ring located closest to the input bevel gear) ii.

Position these components in the speci ed order behind the support of the

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iii. iv.

right bearing (located furthest from the input bevel gear) on the left half of the case Slide the end of the shaft intended for the Oil Pump through these components Slide the following components onto the end of the shaft opposite of the Oil Pump in the following order: right bearing and its retaining ring (note that this bearing is positioned in the custom bearing support machined as part of the left casing half

2. Enclose shafts a. Press left and right halves of casing together to enclose shafts and their components i.

Ensure the inner curvatures of each half form one complete curvature (ie they are on the same side) b. Align centers of fastener holes located along the outer edges of each half c. Using the provided casing fasteners (20), fasten each half of the casing together to form one complete case 3. Prepare input shaft a. Slide the remaining components of the input shaft onto the section of shaft protruding from the case b. Ensure that the self locking retaining ring is rst, followed by the bevel gear, and lastly the retaining ring closest to the end of the shaft 4. Mount gearbox to aero engine a. Align the centers of the fastener holes located on the protruding curvature sections of the case with the centers of the fastener holes on the aero engine b. Using the provided mounting fasteners (4), fasten the gearbox ush to the aero engine

To disassemble the gearbox the above procedure can be carried out in the opposite direction.

5.

Conclusion

In summary, this preliminary report includes a comprehensive theoretical development of the mission statement in addition to a full explanation of the gear, shaft, casing, and fastener analyses complete with visual references and justi cation for all assumptions and reasoning. Dimensions of all components including an exploded view of the gearbox and its Bill of Materials are attached in the appendices.

As a group we are fully con dent that the preliminary design outlined by this report provides a viable solution to the mission statement speci ed by the client, however there are a number of improvements that could be made to this solution. These improvements are discussed below in no particular order.

The overall pro le and size of the gearbox can be minimized further by implementing a shaft layout where gears do not contact each other directly above and below each other. The line of action between shafts can be optimized to achieve the minimum distances between shafts that are speci ed in the mission statement. A simple diagram below shows some arbitrary 1 and 2 used to minimize this distance. This design could not be implemented due to the progress made prior to designing the casing; this design implicitly loads shafts in the xy plane (taking the z axis as analogous

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to changes in altitude) which were not present in our shaft loading calculations. Therefore the nal casing the designed with gears directly above one another.

Additional weight reduction can be obtained via reducing the material volume of the gears. This is not recommended for gears G1, G4, and G5 as they are of smaller diameter and the volume changes may have an e ect of the rim thickness factor which could have adverse e ects on bending stresses.

For gears G2 and G3 of greater diameter however this method could be quite e ective. One potential weight relief volume removal is shown below for G2:

Our current solution includes a dependence on grooves to hold retaining rings that axially locate our components. Although some care was taken to avoid placing these near points of maximum loading,

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they do still cause some critical locations that could be eliminated through the use of self-locking retaining rings that do not required grooves, or the use of sleeves to hold shaft components in place.

The sleeves would further reduce the weight of our design as they would eliminate the need for large shoulder diameters.

In order to accommodate shafts of di erent lengths, it was necessary to design a complex case geometry. A more basic layout with equal length shafts may allow for a casing of smaller volume and subsequently of smaller weight.

In order to reduce the overall assembly and disassembly procedure time it may also be possible to reduce the number of fasteners used to hold the two halves of the case together. The time taken to tighten nuts may also be eliminated by implementing threaded fastener holes that are enclosed at the depth of the screw.

At the end of the day hindsight is 20/20 and these improvements are easily identi able, however things are not always so simple during the conceptual and preliminary design phase. These improvements are included for the purpose of notifying the client of alternative solutions that may be more suitable for their needs.

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Appendix A: Shear, Bending Moment and Torque Diagrams

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Appendix B: Catalogues

B1. Shaft key standard imperial dimensions

B2. Typical Maximum Ranges for Slopes and Transverse De ections

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B3. Weibull Parameters Rating Lives:

B4. Radial Load Factors for Ball Bearings

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B5. SKF Bearing Catalogue

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39

40

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Appendix C: 3D CAD

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43

Shaft B Above, Shaft A Below

Idler Shaft Above, Input shaft Below

Appendix D: Machine Drawings

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