lateral offset gear reducer

Lateral Offset Gear Reducer

Machine Design MCG 3131

Professor: Dr. Ahsan Ahmed

Submitted by Group 22: [Himanshu Sehgal, 8688440]

[Harjot Chahal, 8766203] [Ethan Rodrigues, 300006583] [Rahul Sharma, 300019592]

Due: April 2, 2020

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Abstract

Gear reducers are essentially the intermediary gear train that connects the motor to the machine, its main purpose is the reduction in speed from the input shaft to the output shaft through sequencing of gear. Although they reduce the output speed, the output torque is also magnified by the given factor. When designing a gear reducer it is essential to identify the primary method of motion, power, speed, weight, ambient temperature, operating conditions, life, characteristics of the driven machine, overhung or radial loads, and any space restrictions the gear reducer may have. Although for this analysis we hold many of these criterias to ideal conditions, depending on the context of the gear reducer, other criterias can be considered. The gear reducer will be used for the speed reduction in a cement mixer where the motor speed is too high for the application of mixing cement so it needs to be reduced using a gear train. Given that the input power is 0.3 horsepower and input velocity of 100 revolutions per minute, the reducer must provide an output velocity of 25 revolutions per minute. A two step reduction is used to make the input and output shafts on the gear reducer laterally offset, this was achieved by 3 shafts and 4 helical gears. This report will consist of in-depth shaft, bearing and gear analysis, which will contribute to the overall design of the gear reducer.

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TableofContents

List of Figures 4

List of Tables 5

List of Acronyms 6

Introduction 8

Background and Context 9

3 Design Analysis 10 3.1 Gear Design 10

3.1.1 Gear Specifications 10 3.1.2 Force Analysis 16 3.1.3 Gear Material 17 3.1.4 Gear Wear and Strength Analysis 17

3.2 Shaft Design 21 3.2.1: Shaft Material Selection 21 3.2.2 Shaft 1 (Input Shaft) 21 3.2.3 Shaft 2 (Intermediate Shaft) 27 3.2.4 Shaft 3 (Output Shaft) 33 3.2.5 Safety factor calculations : 39 3.2.6 Shaft Mounting 42

3.3 Bearing Analysis 43 3.3.1 Force Analysis 45 3.3.2 Sizing 45 3.3.3 Mounting and Maintenance 46

4 Conclusion 47

5 References 48

APPENDICES : 49 APPENDIX I: Engineering Drawings 52 APPENDIX II: Juvinall Textbook Tables 62

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List of Figures

Figure 1: Design rendered with ¼ ball bearings cut to show mechanism inside. 8 Figure 2: Design rendered with ¼ ball bearings cut to show mechanism inside. 9 Figure 3: SH6-15 Helical Gear 14 Figure 4: SH6-33 Helical Gear 15 Figure 5: SH6-13 Helical Gear 15 Figure 6: SH6-24 Helical Gear 15 Figure 7: SH6-15 Helical Gear Shaft 22 Figure 8: Input Shaft design 23 Figure 9: Free body diagram in XY plane 23 Figure 10: Free body diagram in XZ plane 24 Figure 11: Shear force diagram 24 Figure 12: Bending moment diagram 25 Figure 13: Deflection diagram 25 Figure 14: Slope diagram 26 Figure 15: Shear stress diagram 26 Figure 16: Bending stress diagram 27 Figure 17: Tabulated results for shaft 1 27 Figure 18: SH6-33 and SH6-13 Helical Gear Intermediary Shaft 28 Figure 19: Intermediate shaft design 29 Figure 20: Free body diagram in XY plane 29 Figure 21: Free body diagram in XZ plane 30 Figure 22: Shear force diagram 30 Figure 23 : Bending moment diagram 31 Figure 24: Deflection diagram 31 Figure 25: Slope diagram 32 Figure 26: Shear stress diagram 32 Figure 27: Bending stress diagram 33 Figure 28: Tabulated results for shaft 2 33 Figure 29: SH6-24 Helical Gear output shaft 34 Figure 30: Output shaft design 34 Figure 31: Free body diagram in XY plane 35 Figure 32: Free body diagram in XZ plane 35 Figure 33: Shear force diagram 36 Figure 34: Bending moment diagram 36 Figure 35: Deflection diagram 37 Figure 36: Slope diagram 37 Figure 37: Shear stress diagram 38 Figure 38: Bending stress diagram 38 Figure 39: Overview of various calculations for output shaft 39 Figure 40: Bearing Placement 44

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Figure 41: Solidworks rendering of Ball Bearing from NTN Catalogue A-1000-XI 44 Figure 42: Equivalent Load Equation 46 Figure 43: HPC Catalogue page 664 50 Figure 44: HPC Catalogue page 665 51 Figure 45: NTN Ball Bearing Catalogue A-1000-XI page 18 52 Figure 46: Final Assembly engineering drawing 53 Figure 47: SH6-15 Helical gear engineering drawing 54 Figure 48: SH6-33 Helical gear engineering drawing 55 Figure 49: SH6-13 Helical gear engineering drawing 56 Figure 50: SH6-24 Helical gear engineering drawing 57 Figure 51: R14 Ball Bearing engineering drawing 58 Figure 52: R16 Ball Bearing engineering drawing 59 Figure 53: R18 Ball Bearing engineering drawing 60 Figure 54: SH6-15 Helical gear input shaft engineering drawing 61 Figure 55: SH6-33 and SH6-13 Helical gears intermediary shaft engineering drawing 62 Figure 56: SH6-24 Helical gear output shaft engineering drawing 63 Figure 57: Representative Bearing Design Lives 64 Figure 58: Application Factors Ka 64

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List of Tables

Table 1: Gear Specifications (calculated and from HPC catalogue) 11 Table 2: Summary of Forces Acting on Gears 16 Table 3: Materials selected for Each Gear and Materials Properties 17 Table 4: Summary of Gear Teeth Surface Fatigue Strength Analysis 19 Table 5: Summary of Gear Teeth Surface Bending Strength Analysis 21 Table 6: Values obtained from table 8.1 in Juvinall & Marshek textbook 40 Table 7: Safety factor values for input, intermediate, and output shaft 42 Table 8: Tabulated bearing forces 45 Table 9: Calculated Bearing Values, Catalogue Number 46

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List of Acronyms

Acronym/Symbols Definition

rpm revolutions per minute

HP horsepower

SF Safety Factor

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1 Introduction

The main objective of this report is to design a lateral offset gear reducer for use in a cement mixer. The motor speed is too high for the application of mixing cement so it needs to be reduced using a gear train. Given an input power of 0.3 HP and input velocity of 100 rpm, the reducer must provide an output velocity of 25 rpm. A configuration with 4 gears and 3 shafts will be used (Figure 1). All the shafts are parallel but have vertical offsets from each other. The report will cover gear, bearing and shaft analyses going over the types used and the forces on them. Additional information on things such as lubrication, seals and casing will also be covered.

For gear selection, a gear type first has to be selected (spur, helical, bevel etc.). Using the required input and output values for speed and power, the number of gear teeth can be calculated. From there, gears can be selected from a catalogue and further values such as pitch diameter can be calculated. The tangential, radial and axial forces can all be calculated using values found in the previous step. Gear material can be chosen based on price and material performance. The bending strength, fatigue and safety factor can all be calculated using equations provided in the textbook.

For shaft analysis, the material will again be selected based on price and performance. The

free body diagram of the shaft allows for the generation of shear and bending moment diagrams. Deflection, slope and stress diagrams can also be generated.

For bearing analysis, a bearing type must first be selected. The two options are journal and

roller element. The choice depends on price, loading and space allowance. After finding the forces applied on the bearings, operating conditions must be estimated. Selecting a reliability and application factor provides enough information to choose a bearing model from a catalogue.

Figure 1: Gear reducer design rendered with ¼ ball bearings cut to show mechanism inside.

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Figure 2: Gear reducer design rendered with ¼ ball bearings cut to show mechanism inside.

2 Background and Context

Gear reducers, also known as speed reducers, are a type of gear train configuration designed primarily for an increase in output torque or a decrease in output speed. They consist of gears, shafts, bearings and attachments. Once the components of the reducer have been selected based on application requirements, the reducer can be assembled. In order to maximize its life, proper installation and maintenance are required [1]. The gears allow for power to be transmitted between a system of connected parts. Gears work based on the principle that meshing gears have the same linear velocity at their contact point. This linear velocity depends on the radius of the gears and their angular velocity. If a gear meshes with one with half its radius, the smaller gear will have double the angular velocity of the first gear. Gears of different sizes and teeth allow for changes in speed and torque. For this report, a lateral offset gear reducer is designed for a cement mixer. The portable ones used at home have a cement bucket supported by a frame with wheels and handles. A motor is attached to the cement bucket and provides power to spin the cement mixer. The gear reducer designed in this report will allow for the mixer to spin at a slower speed when the power provided causes the input gear to rotate at a speed too high for mixing.

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3 Design Analysis

The purpose of the gearbox design is to produce a reduction in speed from 100 rpm at input to 25 rpm at output. As the speed is reduced at the output, the torque at output shaft is higher compared to torque at the input shaft. To achieve the desired speed reduction, two step reduction is used and the gearbox includes 3 shafts and four helical gears. For this lateral offset speed reducer, a safety factor should be around 1.5. Operating conditions for the lateral offset speed reducer being designed are usually going well known therefore a safety factor of about 1.5 is reasonable. Generally, if operating conditions are known, the safety factor does not have to be higher than 1.1-1.5 and this ensures the cost or functionality of the design is not compromised.

3.1 Gear Design

The gearbox consists of four helical gears. According to the design constraints, the reducer must be lateral offset configuration where Gear 1 has speed of 100 rpm and Gear 4 has a output speed of 25 rpm. HPC Catalogue was used for selecting required gears for the design.

3.1.1 Gear Specifications

Input Speed and Output Speed:

ω1 = 100 rpm or

ω1 = 100 X (2π/60) = 10.4719 rad/s

ω4 = 25 rpm or

ω4 = 25𝑋(2𝜋/60) = 2.618 rad/s

Input and Output Torque:

Power = 0.3 hp

Conversion to Watts: (0.3) (746) = 223.8 W

T = (𝑃𝑜𝑤𝑒𝑟(𝑖𝑛𝑊𝑎𝑡𝑡𝑠))/(𝐴𝑛𝑔𝑢𝑙𝑎𝑟𝑆𝑝𝑒𝑒𝑑(𝑖𝑛𝑟𝑎𝑑/𝑠))

T1 =(223.8)/(10.4729) = 21.3694 Nm

and

T4 = (223.8)/(2.618) = 85.4851 Nm

Obtaining gear train value, e:

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𝑒 = 25/100 = 0.25

Number of Teeth:

We have selected 2 stage reduction to achieve desired speed reduction. It is considered a good design practice to have an odd number of teeth on the pinion. So:

Selecting, N1 = 15 teeth.

It is considered a bad design to make the gear teeth ratio a multiple. Therefore:

N2 = 33 teeth

Which means our reduction at this step is 33/15 = 2.2.

Therefore between Gear 1 and Gear we have 1 : 2.2 step down.

Selecting, N3 = 13 teeth.

Now, the step down required is:

[(100)/(25)] − 2.2 = 1.8

Therefore:

N4 = (13) (1.8) ≈ 24 teeth (23.66).

Now we have, N1 = 15 teeth, N2 = 33 teeth, N3 = 13 teeth, and N4 = 24 teeth.

To verify the achieved reduction:

ω1 = 100 rpm

ω1 / N2 = ω2 / N1

So, 100 / 33 = ω2 / 15

ω2 = 45.4545 rpm

Now, ω2 = ω3 = 45.4545 rpm

ω 4 / N3 = ω3 / N4

ω 4 / 13 = 45.4545 / 24

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ω 4 = 24.6212 rpm (acceptable).

Desired output is 25 rpm and we achieved 24.6212 rpm which is acceptable as both values have only 1.52677% difference.

From HPC catalogue, the selected gears have part number SH6-15 with pitch diameter of 2.625 inches, SH6-33 with pitch diameter of 5.775 inches, SH6-13 with pitch diameter of 2.275 inches, and SH6-24 with pitch diameter of 4.200 inches.

Transverse Diametral Pitch (P):

𝑃 = 𝑁/𝑑

Where, N = Number of Teeth, and d = Pitch diameter.

𝑃 = 15/2.625 = 33/5.775 = 13/2.275 = 24/4.2000 = 5.714𝑡𝑒𝑒𝑡ℎ/𝑖𝑛𝑐ℎ𝑒𝑠

Normal Diametral Pitch (N):

𝑁 = 𝑃/𝑐𝑜𝑠(ℎ𝑒𝑙𝑖𝑥𝑎𝑛𝑔𝑙𝑒)

𝑁 = 5.714/𝑐𝑜𝑠(17.45) = 5.9896𝑡𝑒𝑒𝑡ℎ/𝑖𝑛𝑐ℎ𝑒𝑠

Addendum (a):

𝑎 = 1/(𝑃) = 1/5.714 = 0.175𝑖𝑛𝑐ℎ𝑒𝑠

Addendum Circle Radius:

𝐴𝑑𝑑𝑒𝑛𝑑𝑢𝑚𝐶𝑖𝑟𝑐𝑙𝑒𝑅𝑎𝑑𝑖𝑢𝑠 = [𝑅/2] + 𝑎

𝑅 = 𝑑 + 2(𝑎)

For Gear 1:

= [(2.625 + 2(0.175))/2] + 0.175 = 1.6625𝑖𝑛𝑐ℎ𝑒𝑠

Dedendum (b):

𝑏 = 1.25/𝑃 = 1.25/5.714 = 0.2188𝑖𝑛𝑐ℎ𝑒𝑠

Dedendum Circle Radius:

𝐷𝑒𝑑𝑒𝑛𝑑𝑢𝑚𝐶𝑖𝑟𝑐𝑙𝑒𝑅𝑎𝑑𝑖𝑢𝑠 = [𝑅/2] − 𝑏

𝑅 = 𝑑 + 2(𝑎)

For Gear 1:

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𝐷𝑒𝑑𝑒𝑛𝑑𝑢𝑚𝐶𝑖𝑟𝑐𝑙𝑒𝑅𝑎𝑑𝑖𝑢𝑠 = [(2.625 + 2(0.175))/2] − 0.2188 = 1.2687𝑖𝑛𝑐ℎ𝑒𝑠

Base Circle Radius:

𝐵𝑎𝑠𝑒𝐶𝑖𝑟𝑐𝑙𝑒𝑅𝑎𝑑𝑖𝑢𝑠 = [𝑑/2]𝑐𝑜𝑠(𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑎𝑛𝑔𝑙𝑒)

For Gear 1:

=[2.625/2]𝑐𝑜𝑠(20) = 1.2333𝑖𝑛𝑐ℎ𝑒𝑠

Transverse Circular Pitch (Pt):

Pt= π/𝑃

= π/5.714 = 0.5498𝑖𝑛𝑐ℎ𝑒𝑠

Axial Pitch:

𝐴𝑥𝑖𝑎𝑙𝑃𝑖𝑡𝑐ℎ = 𝑃𝑡/𝑐𝑜𝑠(ℎ𝑒𝑙𝑖𝑥𝑎𝑛𝑔𝑙𝑒)

= 0.5498/𝑐𝑜𝑠(17.45) = 0.5763𝑖𝑛𝑐ℎ𝑒𝑠

Normal Circular Pitch:

𝑁𝑜𝑟𝑚𝑎𝑙𝐶𝑖𝑟𝑐𝑢𝑙𝑎𝑟𝑃𝑖𝑡𝑐ℎ = (𝑃𝑡)𝑐𝑜𝑠(ℎ𝑒𝑙𝑖𝑥𝑎𝑛𝑔𝑙𝑒)

= 0.5498𝑐𝑜𝑠(17.45) = 0.5245𝑖𝑛𝑐ℎ𝑒𝑠

Table 1: Gear Specifications (calculated and from HPC catalogue)

Gear Specification Gear 1 Gear 2 Gear 3 Gear 4

Number of Teeth 15 33 13 24

Part Number SH6-15 SH6-33 SH6-13 SH6-24

Transverse Diametral Pitch (P) (teeth/inches) 5.714 5.714 5.714 5.714

Normal Diametral Pitch (N) (teeth/inches) 5.9896 5.9896 5.9896 5.9896

Pitch Diameter (d) (inches) 2.625 5.775 2.275 4.200

Catalogue Bore (inches) 1.000 1.250 0.750 1.000

Boss Diameter (inches) 1.625 2.500 1.625 2.000

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Face Width (inches) 2.000 2.000 2.000 2.000

Pressure Angle (Degrees) 20 20 20 20

Helix Angle (Degrees) 17.45 17.45 17.45 17.45

Addendum (a) (inches) 0.1750 0.1750 0.1750 0.1750

Addendum Circle Radius (inches) 1.6625 3.2375 1.4875 2.4500

Dedendum (b) (inches) 0.2188 0.2188 0.2188 0.2188

Dedendum Circle Radius (inches) 1.2687 2.8437 1.0937 2.0562

Base Circle Radius (inches) 1.2333 2.7134 1.0689 1.9734

Transverse Circular Pitch (Pt) (teeth/inches) 0.5498 0.5498 0.5498 0.5498

Axial Pitch (teeth/inches) 0.5763 0.5763 0.5763 0.5763

Normal Circular Pitch (teeth/inches) 0.5245 0.5245 0.5245 0.5245

The following illustrations dictate the Solidworks renderings of each individual helical gears.

Figure 3: SH6-15 Helical Gear

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3.1.2 Force Analysis

All four gears used in the design are helical gears with pressure angle of 20° and helix angle of 17.45°. There are three major types of forces that act on these helical gears as they mesh. These forces are: Tangential Force, Axial Force, and Radial Force.

Pitch Line Velocity (V):

𝑉 = 𝜋(𝑑)(𝑛)(1/12) Where, d is the Pitch Diameter (inches), and n is the gear rpm. For example for Gear 1 (Pinion):

𝑉 = 𝜋(2.625𝑖𝑛𝑐ℎ𝑒𝑠)(100𝑟𝑝𝑚)(1/12) =68.7223 ft/min

Tangential Forces (Ft): 𝑊 = (𝐹𝑡/𝑉)(33,000)

Ft = (𝑊)(33,000)(1/𝑉) Where, V is the Pitch Line Velocity (ft/minute), W is the power (hp), and Ft is the Tangential Force Component (lbf). For Gear 1 and Gear 2:

Ft= (1/68.7223𝑓𝑡/𝑚𝑖𝑛)(33,000)(0.3) =144.0580 lbf or 1 lbf = 4.4 N

Ft = (144.0580)(4.4) = 633.8554𝑁

Axial Force (Fa): Fa = 𝐹𝑡𝑡𝑎𝑛(ℎ𝑒𝑙𝑖𝑥𝑎𝑛𝑔𝑙𝑒)

For Gear 1 and Gear 2: Fa = 633.8554𝑡𝑎𝑛(17.45) = 199.2459𝑁

Radial Force (Fr):

Fr = 𝐹𝑡𝑡𝑎𝑛(𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑎𝑛𝑔𝑙𝑒) For Gear 1 and Gear 2:

Fr = 633.8554𝑡𝑎𝑛(20) = 230.7045𝑁

Total Force (F): F = [𝐹𝑡/𝑐𝑜𝑠(ℎ𝑒𝑙𝑖𝑥𝑎𝑛𝑔𝑙𝑒)][𝑐𝑜𝑠(𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑎𝑛𝑔𝑙𝑒)]

For Gear 1 and Gear 2: F = [633.8554/𝑐𝑜𝑠(17.45)]𝑐𝑜𝑠(20) = 624.3630𝑁

Table 2: Summary of Forces Acting on Gears Gear Tangential Load

(Ft) (N) Radial Force (Fr) (N)

Axial Force (Fa) (N)

Total Force (F) (N)

Pitch Line Velocity (V) (ft/min)

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1 633.8554 230.7045 199.2459 624.3630 68.7223

2 633.8554 230.7045 199.2459 624.3630 68.7223

3 1609.0188 585.6349 505.7784 1584.9228 27.0724

4 1609.0188 585.6349 505.7784 1584.9228 27.0724

3.1.3 Gear Material

For all helical gears, normalized steel was used. Normalized steel was chosen as material for all gears as it tends to have more predictable behaviour with uniform and fine-grained structure. Moreover, normalized steel is usually cheaper than annealed steel but normalizing also produces a harder and stronger metal than the annealing process. It is considered a good design practice to make meshing gears out of material that have similar or same hardness. For Gears 1 and 2, 1050 Normalized Steel was picked as the material to be used whereas for Gears 3 and 4, 1095 Normalized Steel was picked as Gears 3 and 4 have high torque being applied compared to Gears 1 and 2.

Table 3: Materials selected for Each Gear and Materials Properties. Gear Number 1 2 3 4

Material 1050 Normalized Steel

1050 Normalized Steel



Tensile Strength (MPa)

748.1 748.1 1013.5 1013.5

Yield Strength (MPa) 427.5 427.5 499.9 499.9

Brinell Hardness 217 217 293 293

3.1.4 Gear Wear and Strength Analysis

For known operating conditions, safety factors of about 1.1 to 1.5 are required but higher safety factors are also acceptable. It is crucial to know surface fatigue, bending stress and endurance limits for the gears being used.

Gear Tooth Surface Fatigue Analysis: Using Equation 16.13 from the Juvinall and Marshek textbook, Surface fatigue stress (σH):

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Where, Ft is the Tangential Force, b is the Face Width, Km is the Mounting Factor, Kv is the Velocity Factor, Ko is the Overload Correction Factor, Cp is Value of the Elastic Coefficient.

● The gears are ground gears and high precision shaved. Using Fig 15.24 in the Juvinall and Marshek textbook, KV1= KV2 = 1.11 and KV3 = KV4 = 1.07.

● The power source specified is uniform and we have uniform driven machinery. Using Table 15.1 in the Juvinall and Marshek textbook, Ko = 1 for all gears.

● Accurate mountings, small bearing clearances, minimum deflection, precision gears. Using Table 15.2 in Juvinall and Marshek textbook, Km= 1.3 for all gears.

● Pinion Material is steel. Using Table 15.4b in Juvinall and Marshek textbook, the Cp= 191

√MPa. Using Equation 15.9 of the Juvinall and Marshek textbook, CR (Contact Ratio) is:

Where, pb is the Base Pitch, c is the center distance, rb is the Base Circle Radius of meshing gears, ra is the Addendum Circle Radius of meshing gears. Using Equation 15.23 of the Juvinall and Marshek textbook, Geometry Factor (I) is:

𝑅 = 𝑃𝑖𝑡𝑐ℎ𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟𝑜𝑓𝐺𝑒𝑎𝑟𝐵/𝑃𝑖𝑡𝑐ℎ𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟𝑜𝑓𝐺𝑒𝑎𝑟𝐴

Safety factor (n) is: 𝑛 =SH/σH

Where, according to Equation 15.25 of the Juvinall and Marshek textbook, Surface Endurance Limit (SH) is:

SH = Sfe CLi CR Where, Sfe (Surface Fatigue Strength), CLi (Life Factor), CR (Reliability Factor).

● Using Figure 15.27 of the Juvinall and Marshek textbook, CLi= 1.0 for all gears. ● Using Table 15.6 of the Juvinall and Marshek textbook, CR= 0.8 for 99.9% reliability. ● According to class notes, Sfe = 2.8 (Bhn) - 69. ● Using Appendix C-4a of Juvinall and Marshek textbook, Brinell Hardness is 217 Bhn for

1050 Normalized Steel used for Gears 1 and 2. ● Using Appendix C-4a of Juvinall and Marshek textbook, Brinell Hardness is 293 Bhn for

1095 Normalized Steel used for Gears 3 and 4. Sample Calculations for Gear Tooth Surface Fatigue Analysis: Contact Ratio (CR):

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= 1.65

Geometry Factor (I):

= = 0.1105

Surface Fatigue Stress (σH):

= = 224.6333 MPa

Note: inches were converted to mm to obtain an answer in MPa. Surface Endurance Limit (SH):

= [2.8(217) − 69](1)(0.8) = 430.88 MPa

Safety Factor (n): = 430.88/224.6333

= 1.9

Table 4: Summary of Gear Teeth Surface Fatigue Strength Analysis.

Gear Geometry Factor (I)

Contact Ratio (CR)

Surface Fatigue Stress (σH) (MPa)

Surface Fatigue Stress (SH)

Safety factor (n)

1 0.1105 1.65 224.6333 430.88 1.9

2 0.1105 1.65 151.4477 430.88 2.8

3 0.1042 1.47 411.8079 601.12 1.5

4 0.1042 1.47 303.0824 601.12 2.0

Gear Tooth Surface Bending Strength Analysis: Using Equation 16.12 of the the Juvinall and Marshek textbook, Bending Stress (σ) is:

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Where, Ft is the Tangential Force, KV is the Dynamic Factor, Ko is the Overload Factor, Km is the Mounting Factor, J is the gear geometry factor, b is the face width, and P is the Transverse Diametral Pitch.

● Using Figure 16.8 of the Juvinall and Marshek textbook, for J1 = 0.42 (for Gear 1), J2 = 0.52 (for Gear 2), J3 = 0.38 (for Gear 3), and J4 = 0.48 (for Gear 4).

● Accurate mountings, small bearing clearances, minimum deflection, precision gears. Using Table 15.2 in Juvinall and Marshek textbook, Km= 1.3 for all gears.

● The power source specified is uniform and we have uniform driven machinery. Using Table 15.1 in the Juvinall and Marshek textbook, Ko = 1.25 (assuming moderate shock for safety) for all gears.

● The gears are ground gears and high precision shaved. Using Fig 15.24 in the Juvinall and Marshek textbook, KV1= KV2 = 1.11 and KV3 = KV4 = 1.07.

According to Equation 15.18 of the Juvinall and Marshek textbook, Bending Fatigue Strength (Sn) is:

Where, S’n is the standard R.R Moore Endurance Limit (0.5 Su), CL is the Load Factor, CG is the Gradient Factor, kr is the Reliability Factor, kt is the Temperature Factor, kms is the Mean Stress Factor.

● Using Table 8.1 of the Juvinall and Marshek textbook, CL = 1 for all gears. ● Using Table 8.1 of the Juvinall and Marshek textbook, CG = 1 for all gears. ● Using Figure 8.13 of the Juvinall and Marshek textbook, CS = 0.9 (all gears) for fine-

grounded or commercially polished surfaces. ● Using Table 15.3 of the Juvinall and Marshek textbook, kr = 0.659 (all gears) for 99.999%

reliability. ● According to the Equation 15.19 of the Juvinall and Marshek textbook, kt = 1 (for T > 160

F) and kms = 1.4 (for non-idler / input and gears). Safety Factor (SF):

𝑆𝐹 = 𝑆𝑛/𝜎

Table 5: Summary of Gear Teeth Surface Bending Strength Analysis Gear Bending Stress (σ) (MPa) Bending Strength (Sn) (MPa) Safety Factor (SN)

1 11.2115 310.5887 27.7

2 9.0555 310.5887 34.3

3 30.3223 420.7748 13.9

4 24.0052 420.7748 17.5

Sample Calculations for Gear Tooth Surface Bending Strength Analysis: Bending Stress (σ):

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= 11.2115 MPa Bending Strength (Sn):

(0.5)(748.1)(1)(1)(0.9)(0.659)(1)(1.4) =310.5887 MPa Safety Factor (SF):

310.5887/11.2115= 27.7

3.2 Shaft Design

Based on the knowledge obtained in class, it is known that elements such as gears and belts induce axial, transverse and torsional loads on the shaft. A shaft should be designed to be as short, stiff and light as possible to limit high bending loads and to be as efficient as possible. In order to reduce shaft deflection and slope, the gear mesh separation is needed to be very small. In addition, for torsional deflection, the permissible angle of twist should be less than 0.08º per foot for machined shafts [Class notes, Shafting]. Also any steps in the shaft should be at a maximum of 20% increments and have a small fillet if permissible. Finally, it must be configured such that stress concentration sites do not interfere with high stress regions.

All the components of the shaft will be analysed to see how they will be mounting to the

shafts in terms of thermal contraction and expansion, keys, splines and setscrews. A force analysis will be conducted; considering the point of contact between the gears and shaft and therefore the different loads that are applied on the shaft, reaction forces, and the bearings needed to balance the forces.

3.2.1: Shaft Material Selection

Based on the torque calculations as well as the gear calculations, it is evident there is a significant amount of torque being applied which also means large reaction forces. when selecting the input and output shafts, a material needed to be picked which could handle the high torque and force values so it was determined that AISI 1020 steel would be used. As for the shaft supporting gear 2 and gear 3 will be made of AISI 1095 steel to ensure it can handle the torque and will be stiff enough to resist buckling or twisting.

3.2.2 Shaft 1 (Input Shaft)

The input shaft is made of AISI 1020 steel which will have a machined finish. The shaft will have the pinion gear (gear 1) mounted on it by means of a key and will also have 2 fixed - floating bearings mounted to support it. The shaft is designed to have multiple steps to help support the items being mounted as well. The start of the shaft has a diameter of 1.00in which is where Bearing A will be mounted. The bearing will be mounted up against the step to make sure it does not move. The first step goes from 1.00in to 1.125in in diameter and this is to help support the bearing and the pinion gear. After the first step is another step down to 1.00in again and this is where the pinion gear is mounted by a key. Following this portion of the shaft is another step down

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where Bearing B is mounted. The shaft diameter goes from 1.00in to 0.875in so that the bearing can be placed up against the step and be held in place on one side.

Figure 7: SH6-15 Helical Gear Shaft

Figure 8: Input Shaft design

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Figure 9: Free body diagram in XY plane

Figure 10: Free body diagram in XZ plane

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Figure 11: Shear force diagram

Figure 12: Bending moment diagram

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Figure 13: Deflection diagram

Figure 14: Slope diagram

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Figure 15: Shear stress diagram

Figure 16: Bending stress diagram

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Figure 17: Tabulated results for shaft 1

3.2.3 Shaft 2 (Intermediate Shaft)

The intermediate shaft is made of AISI 1095 steel. This shaft is the biggest shaft in our design at 7.85in as it has two gears and 2 bearings mounted on it. Similar to the input shaft, the gears are mounted using keys and the bearing are mounted in a fixed- floating configuration. The shaft starts with a diameter of 1.125in and has bearing C mounted on it up against the step. The first step increases the diameter from 1.125in to 1.375in for the sole purpose of providing support for the bearing and gear. A step down is done again to reduce the diameter to 1.250in to mount gear 2. After gear 2 is mounted there is another step down to make the diameter 1.00in and this is to support bearing d. The bore diameter of bearing d is 0.875in so another step down was made. After the step down for the mounting of bearing d was yet another step down to mount gear 3. The bore diameter for gear 3 was 0.750in so the diameter was reduced from 0.875in to 0.750in.

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Figure 18: SH6-33 and SH6-13 Helical Gear Intermediary Shaft

Figure 19: Intermediate shaft design

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30



31



32



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Figure 28: Tabulated results for shaft 2

3.2.4 Shaft 3 (Output Shaft)

The output shaft is the last shaft in our design. It is made of the same material as shaft 1 which was AISI 1040. This shaft will have the last gear (gear 4) mounted on it by means of a key and will also be supported by 2 bearings in a fixed- floating position. The output shaft has an identical design to the input shaft in terms of placements of the gear and bearing as well as the diameters and lengths.

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Figure 29: SH6-24 Helical Gear output shaft

Figure 30: Output shaft design

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36



37



38



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Figure 39: Overview of various calculations for output shaft

3.2.5 Safety factor calculations :

To confirm that that design of the shafts as well as the material selection is appropriate for our tasks, a safety factor will be calculated for the 3 shafts. To calculate the safety factors for the shafts, the modified goodman equation will be used.

To show our calculations, sample calculations will be shown for the 2nd shaft as it is the most complex. The first step taken was to solve for the endurance limit Sn by using equations and values available in table 8.1 in the textbook by Juvinall and Marshek. Table 6: Values obtained from table 8.1 in Juvinall & Marshek textbook

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Bending Axial Torsion

CL 1.0 1.0 0.58

CG 0.9 0.8 0.9

CS 0.8 0.8 0.8

CT 1.0 1.0 1.0

CR 1.0 1.0 1.0

The next step taken was to solve for the fatigue concentration factor Kf at the various steps. This was done by using the following equation found in the textbook:

where q and Kt are found by using figure 8.24 & figure 4.35 found in the textbook For step 1:

For step 2:

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For step 3:

For step 4:

Based on the fatigue concentration factors, step one has the highest concentrations so it will be used to compute the safety factor. The next step is to determine the bending and axial stress as well as the torsion. The following equations can be found in the textbook:

The general biaxial formulas found in table 8.2 in the textbook are used to find the stress values to be used in the modified goodman equation.

Now that all the variables required to calculate the safety factor have been solved for, the safety factor can be calculated for bending, axial, and torsion using the modified goodman equation:

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The safety factors for the input and output shaft are also calculated using the same approach and the values are tabulated in table 7. Table 7: Safety factor values for input, intermediate, and output shaft

Safety Factor Input Intermediate Output

nbending 12.60 2.08 4.39

naxial 11.48 1.86 4.05

ntorsion 8.05 1.22 2.93

3.2.6 Shaft Mounting

To mount the gears onto the shaft, keys and keyways are being used. keys were chosen because they are good at handling medium to high torque and are fairly cheap when compared to other components in the gear reducer. In a worse case scenario, it is always preferred that the key fail before anything else because they can be easily replaced and do not cost a lot compared to a bearing or helical gear failing. The key design that we chose to use was the pratt and whitney design as it is a simple rectangular design.

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3.3 Bearing Analysis

Bearings are components designed to transmit loads at their contact surfaces. When attached to a rotating shaft, there will be relative motion between the contacting surfaces. It is important to minimize friction and wear caused by this motion, which can be done by a lubricant. The two types of bearings are roller element and journal. Since the reducer uses helical gears, there will be axial as well as radial forces. Journal bearings would be able to take higher loads than roller element bearings but would require a constant lubrication system. In order to minimize space and cost, radial ball bearings will be used. Roller element bearings are also better suited to cold environments [2], which the cement mixer may need to operate under. The bearings will be placed in the configuration below (Figure 40). This placement allows for axial and radial forces to be supported in all directions. A CAD model of the bearing can be seen in Figure 41.

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Figure 40: Bearing Placement

Figure 41: Solidworks rendering of Ball Bearing used in the assembly, R14, R16 and R18 from NTN Catalogue A-1000-XI

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3.3.1 Force Analysis

The forces that will be transmitted to the bearings come from the gears. The bearings will be loaded with both axial and radial forces. The bearings and forces are listed in Table 7. Table 8: Tabulated bearing forces

Bearing Radial Force (Fr) (N)

Axial Force (Fa) (N)

Fa / Fr

1 163.146 143.645 0.88

2 67.289 55.601 0.83

3 81.706 158.919 1.94

4 848.045 546.106 0.64

5 414.825 364.637 0.88

6 170.81 141.141 0.83

3.3.2 Sizing

In order to choose a bearing model, operating conditions need to be estimated. The shaft speeds for shafts 1, 2 and 3 are ω1 = 100 rpm, ω2 = 45.4545 rpm and ω3 = 24.6212 rpm. An 8 hour per day, 5 day per week operation with constant radial and axial loads will be assumed. Using the operation hours, the design life of 30,000 hours can be selected from table 14.4 in the Juvinall textbook.

From table 14.3 in the Juvinall textbook, an application factor Ka of 1.5 will be chosen. This value corresponds to a light to moderate impact for shock loading. The loadings on the bearings should be constant, but it is possible that a piece of unhardened cement or debris could cause the machine to stop suddenly, causing shock loading on the bearings. A standard reliability of 90% will be chosen, so the reliability factor Kr = 1. Using equation 14.3 (Figure 42) in the Juvinall textbook, equivalent loads can be calculated. The design life in revolutions can be calculated by multiplying the shaft speed by the design life in minutes. Using equation 14.5b in the Juvinall textbook, the bearing rated capacity can be found. The NTN Ball Bearing Catalogue A-1000-XI will be used to select bearing models. The selected bearing models are rated for loads at least 3 times the calculated bearing rated capacities. A summary of calculated bearing values, catalogue numbers and dimensions can be seen in Table 8.

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Figure 42: Equivalent Load Equation Table 9: Calculated Bearing Values, Catalogue Number Bearing Equivalent

Load Fe (N)

Design Life (Revolutions)

Bearing Rated Capacity (N)

Bearing Catalogue Number

Bore (Inches)

Outer Diameter (Inches)

Width (Inches)

1 191.6 1.8E+8 479 R16 1 2 0.5

2 75.8537 1.8E+8 190 R14 0.875 1.875 0.5

3 171.6283 8.18181E+7 331 R18 1.125 2.125 0.5

4 848.045 8.18181E+7 1640 R14 0.875 1.875 0.5

5 486.6487 4.431816E+7

800 R16 1 2 0.5

6 192.5515 4.431816E+7

317 R14 0.875 1.875 0.5

3.3.3 Mounting and Maintenance

The bearings will be mounted onto the shaft using steps. If the bearings are pre lubricated then they don’t need to be cleaned before mounting. If not, they should be cleaned with filtered oil and provided with corrosion protection. Shielded or sealed models can be ordered to prevent contamination from debris. For mounting, a press fit can be done using a mounting tool. It’s a good idea to lubricate the shaft before mounting the bearings [3].

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4 Conclusion In conclusion the report goes in an in-depth analysis of the design of a lateral offset gear

reducer which was used in a cement mixer. In the application of a cement mixer the input motor speed is relatively high for its application of mixing cement so it is essential for the reduction in the speed of the motor through the gear reducer. Given an input power of 0.3 HP and input velocity of 100 rpm, the reducer provides an output velocity of 25 rpm. A configuration with 4 gears and 3 shafts was used as seen in figure 1. The approach to solving the project had a number of iterations, various designs were explored to achieve the final product design. The assembly consists of 202 components (components of sub-assemblies included), including the three different types of ball bearings (the assembly consisted of repeating parts, such as multiple bearings, rivets, internal bearing balls, ball cage, etc.) used, three different shafts (input, intermediary, output), and four helical gears. The gear and bearing were chosen from the HPC and NTN A-1000-XI catalogues, the specifications based from the catalogues were used to conduct a gear analysis. The gear analysis consisted of the calculations for the input and output speed of each gear, gear sizing calculations (number of teeth, transverse diametral pitch, etc.), force analysis (ex. Pitch line velocity, tangential forces, axial, and radial forces, etc.), material selection based on each materials' properties, and wear and strength analysis of the gears (ex. Safety factor, surface fatigue stress, surface endurance limit.

Once the gear selection was completed from the HPC catalogue, the shaft design was implemented, and a shaft analysis was conducted, elements in the gear reducer such as the helical gears induce axial, transverse and torsional loads on the shaft. The shaft that was designed was as short, stiff and light as possible to limit high bending loads and to further increase efficiency. The calculated torque was used to determine the material the shaft would be made from, and the gear bore diameters outlined the final shaft configuration. A force analysis on the shaft was conducted (ex. free body diagrams, bending moment, deflection, shear stress, etc.) for each shaft, and its results tabulated. Critical points on the shaft such as keyways, and keys were analyzed to their effects of the overall shaft performance and strength. Based on these calculations a factor of safety is determined for each shaft.

Bearings are components designed to transmit loads at their contact surfaces and when attached to a rotating shaft, there will be relative motion between the contacting surfaces. It was essential to minimize the friction and wear caused by this motion, this was achieved through the use of a lubricant. Since the design for the gear reducer relied on helical gears (these induced axial and radial forces), out of the two types of bearings under consideration (roller and journal), the roller bearings were selected. Journal bearings would be able to take higher loads than roller element bearings but would require a constant lubrication system and in order to minimize space and cost, angular contact ball bearings were used. A force analysis was conducted on the bearing and the NTN A-1000-XI catalogue was used to locate bearings required bearings (R14,R16, and R18 bearings were used from the catalogue). With all components in correlation, the lateral offset gear reducer design was achieved.

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5 References

[1] C. Popp, “Speed Reducers,” Machine Design. [Online]. Available:

https://www.machinedesign.com/automation-iiot/article/21827591/speed-reducers. [Accessed: 26-Mar-2020].

[2] “Journal Bearings vs. Ball Bearings - What's the Difference?,” Thomas. [Online].

Available: https://www.thomasnet.com/articles/machinery-tools-supplies/journal-bearings-vs-ball-bearings/. [Accessed: 26-Mar-2020].

[3] “Maintenance & Repairs,” 2. Mounting | Maintenance & Repairs | Services | NSK

Global. [Online]. Available: https://www.nsk.com/services/maintenancerepairs/mounting.html. [Accessed: 30-Mar-2020].

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APPENDICES :

Figure 43: HPC Catalogue page 664

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Figure 44: HPC Catalogue page 665

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Figure 45: NTN Ball Bearing Catalogue A-1000-XI page 18

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APPENDIX I: Engineering Drawings

Figure 46: Final Assembly engineering drawing

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Figure 47: SH6-15 Helical gear engineering drawing

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Figure 51: R14 Ball Bearing engineering drawing

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Figure 54: SH6-15 Helical gear input shaft engineering drawing

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Figure 55: SH6-33 and SH6-13 Helical gears intermediary shaft engineering drawing

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Figure 56: SH6-24 Helical gear output shaft engineering drawing

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APPENDIX II: Juvinall Textbook Tables

Figure 57: Representative Bearing Design Lives

Figure 58: Application Factors Ka

lateral offset gear reducer

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