Introduction. Design Specifications

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Austen Robbins
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1 Executive Summary This project entails the optimization of a gearbox. The system s requirements are up to the designer and the specified use is for a 0 hp Briggs & Stratton engine, commonly used gokarts. With 5500 rpm as a maximum, a two stage compound gearbox will be utilized in order to minimize the size of the system. Using Shigley s Mechanical Engineering Design text, theoretical calculations yielded that three separate shafts with a 20 tooth gear mated to a 35 tooth gear, on each stage, will provide the necessary reduction in engine speed to accommodate the 3: overall gear train ratio. The required shafts are 0 mm in diameter and made of 22 Hot Rolled Steel. To fix the gears to the shafts, 2.4 mm 020 Cold Drawn Steel keys will be utilized. Also, 0 mm bore two-piece locking collars will be used to ensure the gears do not move axially. 0 mm bore flanged double sealed ball bearings are pressed into the gearbox casing to allow smooth rotation of the assembly, without allowing dirt or other contaminants to hinder the bearing s function.

2 Table of Contents Introduction...4 Design Specifications...4 Design Geometry...5 Design Analysis / Discussion...6 Conclusion...9 References...0 Appendix A: Gearbox Components... Appendix B: Mathcad Calculations...3

Introduction Many machines have a motor that operates at a high rotational speed. In order to make use of this high rotational speed, a gearbox is needed.

3 Introduction Many machines have a motor that operates at a high rotational speed. In order to make use of this high rotational speed, a gearbox is needed. By combining a motor and a gearbox, the rotational speed from the motor can be reduced, thus creating useful torque. This project examines a simple gearbox with one input shaft, one output shaft, and one internal shaft. Figure displays the system, as well as the appropriate bearings, collars, and spacers to make the gearbox operable. Coupling the use of I-DEAS with Mathcad allows theoretical and actual stresses to be compared. This will be discussed in the Design Analysis section. With this exposition completed, it is appropriate to look at the Design Specifications for the gearbox. Design Specifications Figure : Gearbox This gearbox will be utilized in an SAE Mini Baja vehicle. While originally, the gearbox was going to act as a gear in an automobile transmission, due to the gears available and the desire to keep the gearbox simple, an automobile transferred too much torque to reduce the speed with only 4 gears. Thus, the design was altered to be more realistic. The following specifications are a summary of the design. Power Output: 0 hp

4 Max RPM (assuming over-rev): 5500 rpm Overall Gear Ratio: 3: Number of Gears: 4 o 2-20 tooth, 22 mm Ground Steel Spur Gears o 2-35 tooth, 37 mm Ground Steel Spur Gears 3-0 mm Diameter, 22 Steel Hot Rolled Shafts mm, 020 Steel Cold Drawn Keys 4-0 mm Bore, Two-Piece Shaft Collars 6-0mm Bore, Flanged Double Sealed Ball Bearings 27 mm Case height, 0.6 mm wall thickness Appendix A contains the data sheets for these components, as this gearbox utilizes off-the-shelf products. With the Design Specification completed, it is necessary to observe the geometry of the components that comprise the gearbox. Design Geometry With the components selected, the orientation of these components is up to the designer. Some gearboxes are compound reverted, where the input and output shafts in-line, while others have the input and output shafts in succession. To simplify the design, it was decided that the gearbox would be a two stage compound system, with three separate shafts. Due to the fact that the mathematical analysis was completed prior to the CAD drawing, the overall dimensions of the gearbox were known. Figure 2 shows the gearbox from a top view, allowing all of the components to be visible.

5 Figure 2: Top View of Gearbox To keep the gears from sliding laterally, a keyway is machined on the shafts to mate with the keyway on the key. While some gears utilize set screws, a key is preferred as it is more appropriate for high powered machine. Furthermore, two-piece collars are used on gears to ensure no movement. To secure the shafts into the gearbox casing, double sealed single flange ball bearings are pressed into the casing. With the Design Geometry depicted, it is feasible to look at the Design Analysis. Design Analysis / Discussion When analyzing the system, Shigley s text is a necessity. The case study presents a clear methodology to analyze a power transmission and while it is extensive, it is necessary. With this being said, Appendix B contains the entire workbook (completed in Mathcad), that corresponds to every component in the gearbox. While there a plethora of equations used to size the entire gearbox, there a select few which are worth denoting. Equation () allows the torques in each shaft to be calculated, while Equation (2) allows the rotational speed of each subsequent shaft to be calculated. ()

6 Where T is the torque in N-m, is the power in Watt, and is the speed of the shaft in rpm. (2) Where is the subsequent shaft speed, in rpm, and is the ratio between gears. By assuming a size for the gearbox casing, it is possible to determine the minimum diametral pitch of the gears,, which has units teeth/mm. Using the diametral pitch, Equation (3) allows the diameter of the gears to be computed. (3) Where is the diameter of the gear in mm and which is the number of teeth on the given gear. Using the values from Equations () - (3), it is appropriate to compute the pitch line velocity and ultimately, the transmitted load between gears. Equation (4) allows the pitch line velocity,, to be computed in mm/s, while Equation (5) solves for the transmitted load,, to be computed in N. And (4) These 4 equations are the basis for the entire design. Completing this analysis displays certain information, regarding the components, which allows the analysis in I-DEAS to be simplified. (5)

7 This includes the fact that gear 3 is the smallest gear on the intermediate shaft, yet transmits the largest load. Thus, it is only necessary to analyze that gear due to the fact that if it can handle the design requirements, than all other gears are able to as well. This same ideology is applied to the shafts. The intermediate shaft was designed for since gear 3 has the largest bending moment, since gear 3 is smaller than gear 2, yet must transmit the same torque. While this diameter most probably could be reduced for the input and output shafts, all shafts were assumed to be the same diameter in an attempt to simplify the gearbox. It is now appropriate to look at the comparison between the FEA results and the theoretical calculations for the gear, the shaft, and the key. Using the transmitted load on gear 3, as calculated in Appendix B, the FEA yielded a stress of 75,000 psi, as opposed to the machine design calculations, which yielded a stress of 26,000 psi. The FEA results can be seen in Figure 3. Figure 3: Gear 3 FEA While this has a discrepancy of 40,000 psi, it is affirming that the results are reasonably close, as the magnitude between them is less than 2. This discrepancy could be due to the fact that I- DEAS isn t truly applying the force in a realistic manner. Note the stress concentration surrounding the keyway and the deformation of the gear tooth. While this analysis is beneficial in allowing visual results, the way the analysis is performed isn t completely realistic. This is due to the fact that the gear is part of an assembly and will already be rotating on the shaft. Although the force applied is congruous, it won t be acting on a static gear. Moving through the gearbox, the next component that is analyzed is the shaft. In order to analyze the shaft, the location of the gears is noted, and then two boxes are built off the keyway

in order to apply the necessary transmitted loads, as well as the reactions at the bearings. Figure 4 displays the stress analysis on the intermediate shaft.

Once again because the difference in magnitude is about 2.5, this is acceptable.

8 in order to apply the necessary transmitted loads, as well as the reactions at the bearings. Figure 4 displays the stress analysis on the intermediate shaft. Figure 4: Shaft FEA Completing the FEA yielded a stress of 4,000 psi, as opposed to 7,700 psi when using theoretical calculations. Once again because the difference in magnitude is about 2.5, this is acceptable. The last component that is analyzed in the gearbox is the key, which fixes the gears to the shaft. Figure 5 displays the FEA analysis. In comparing the results, FEA yielded a stress of 40,000 psi, while theoretical calculations yielded a stress of 32,000 psi. These results have the minimum discrepancy in magnitude. Figure 5: Key FEA

9 By comparing the FEA results and the principles portrayed in Shigley s text, it is affirming that the values are relatively close to one another. FEA always yields higher stress values due to the fact that FEA is dependent on the meshing and other user-set parameters. Having this data and the presented evaluation completed, a conclusion of the knowledge gained is appropriate. Conclusion This project displays that designing a gearbox is very dependent on the power and rotational speed of the engine it is mated to. In order to minimize the size of the gearbox, it is advisable to use constant proportions (the square root of the overall gear ratio) for each stage. By analyzing the worst case scenario for the gears and shafts, with respect to torque and gear size, all subsequent gears and shafts would be appropriate for the design as well. When comparing the FEA results with those from theoretical calculations, it is inevitable to have discrepancies, but the order of magnitude must be reasonable (ideally less than 3). By using I- DEAS, I know I learned the program a lot more. Although I was a little concerned in the beginning of the project, due to the fact my previous endeavors proved to be futile with it, I am much more confident with the program. Lastly, this project coupled Computer Aided Design with principles learned previously in Machine Design, making for an appropriate fusion of material and an overall enjoyable case study. References ) Budynas, Richard G., J. Keith. Nisbett, and Joseph Edward. Shigley. Shigley's Mechanical Engineering Design. Boston: McGraw-Hill, Print. 2) Walker, Graham. "Gearbox Project." Interview. Print.

10 Appendix A: Gearbox Components Figure A-: 20 Tooth Gear Figure A-2: 35 Tooth Gear

11 Figure A-3: 0 mm Full Keyed Shafts Figure A-4: 2.4 mm Keys Figure A-5: 0 mm I.D. Two-Piece Shaft Collars Figure A-6: 0 mm I.D. Flanged Double Sealed Ball Bearings

12 Analyzed for 2 nd gear, 3: Pmax 0 hp max 5500rpm n 3 n n This is the ratio between each gear. It is now appropriate to solve for the torque in each shaft Pmax T T 9.549ft lbf max max 2 n rpm max 3 n rpm T2 T 2 T2 6.54ft lbf 3 Tout T max Tout ft lbf 3 Appendix B: Mathcad Calculations Solving for the number of teeth for each gear, noting that the teeth on gears and 3 are the same, as well as the teeth on gears 2 and 4, yields: 20deg m n k.43 N 2k ( 2m) ( sin( )) 2 m m 2 ( 2m) ( sin( )) 2 N Thus N 20 N2 Nn

13 N Thus N2 35 N = N3 = 20 teeth N2 = N4 = 35 teeth Force Analysis Starting with an estimate for the overall size of the gear box as 5 inches and wall thickness of.4 inches: Y 5in W.4in N N2 N Pmin Pmin Y W in Thus P 23 in N d d 0.87in P N2 d2 d2.522in P Gear -2 Which corresponds to the minimum diametral pitch Based upon available gears from QTC Gears Thus these diameters will be used to analyze the gearbox, noting that d = d3 and d2 = d4 d V2 2 max V ft min Pmax Wt2 Wt lbf V2 This is the pitch line velocity from gear to 2 This is the transmitted load from gear to 2 Similarly: Gear 3-4 d2 V ft V min Pmax Wt34 Wt lbf V34 This is the pitch line velocity from gear 3 to 4 This is the transmitted load from gear 3 to 4

14 Because gear 3 has the largest transmitted load, it is appropriate to start the analysis with gear 3. Gear 3 Contact Stress and Bending Surface-Strength Geometry Factor cos ( ) sin( ) N I N2 N Dynamic Factor, Kv Qv 7 N2 (Assumed as average quality number) 2 3 B.23( 2 Qv) B A 50 56( B) A B A 27.5 Kv Kv.56 A Face Width F 3 F 0.4in P Load Distribution Factor, Km Congruous with gear spec's. Cmc Cpm Cma.5 For uncrowned teeth For straddle mounted For commercially enclosed unit Ce Cpf F 0 d For gearing not adjusted at assembly F in Cpf 0.05 Km [ Cmc[ ( Cpf Cpm) ( CmaCe )]] Km.65 Size Factor Ks

15 Ko Cf Elastic Coefficient Cp 650 For steel gear on a steel pinion Now to find the contact stress: c Cppsi Wt34 Km Cf KoKvKs lbf d I F in 2 c psi Now it is appropriate to look at the life expectancy for the gear and allowable contact stress L4 5000hr 2 L rev From Figure 4-5: Zn Kr Kt Ch Assume factor of safety of.2: Sh.2 Shc Sc Sc psi Zn From Figure 4-5: Sc psi For Grade 3 Carburized and Hardened ScZn nc nc.29 c Factor. of Safety for wear on gear 3 =.3 With the contact stress found, it is appropriate to look at the bending effect on the gear From Figure 4-6: J.27 Kb

16 P Km Wt34 Kv F J psi From Figure 4-4: Yn From Table 4-3 St 65000psi all StYn all psi all n n.304 Thus bending isn't an issue on the gear. Gear 4 Contact Stress and Bending Same parameters as gear 3, except J, Yn, and Zn: J.4 L hr 3 L rev Yn.97 Zn c Cppsi Wt34 Km Cf KoKvKs lbf d2 I F in 2 c psi P Km Wt34 Kv F J psi Sc psi

17 St 32000psi nc ScZn c.706 n StYn.537 Choose Grade Through Hardened as well. Gear Contact Stress and Bending Surface-Strength Geometry Factor cos ( ) sin( ) N I N2 N Dynamic Factor, Kv Qv 7 N2 (Assumed as average quality number) 2 3 B.23( 2 Qv) B A 50 56( B) A B A 223 Kv Kv.32 A Face Width F 3 F 0.4in P Load Distribution Factor, Km Congruous with gear specs Cmc Cpm Cma.5 For uncrowned teeth For straddle mounted For commercially enclosed unit Ce Cpf F 0 d For gearing not adjusted at assembly F in Cpf 0.05

18 Km [ Cmc[ ( Cpf Cpm) ( CmaCe )]] Km.65 Size Factor Ks Ko Cf Elastic Coefficient Cp 650 For steel gear on a steel pinion Now to find the contact stress: c Cppsi Wt2 Km Cf KoKvKs lbf d I F in 2 c psi Now it is appropriate to look at the life expectancy for the gear and allowable contact stress L4 5000hr max L rev From Figure 4-5: Zn Kr Kt Ch Assume factor of safety of.2: Sh.2 Shc Sc Sc psi Zn From Figure 4-5: Sc psi For Grade 3 Carburized and Hardened

19 ScZn nc nc.58 c Factor. of Safety for wear on gear 3 =.6 With the contact stress found, it is appropriate to look at the bending effect on the gear From Figure 4-6: J.27 Kb P Km Wt2 Kv F J psi From Figure 4-4: Yn.88 From Table 4-3 St 45000psi all StYn all psi all n n.54 Thus bending isn't an issue on the gear. Gear 2 Contact Stress and Bending J.4 Yn.9 Zn c Cppsi Wt2 Km Cf KoKvKs lbf d2 I F in 2 c psi

20 P Km Wt2 Kv F J psi Sc 90000psi St 28000psi nc ScZn c.444 all StYn all psi all n Gear Summary: Gear :.866" diameter,.394" face width, 23 teeth/in Gear 2:.457" diameter,.394" face width, 23 teeth/in Gear 3:.866" diameter,.394" face width, 23 teeth/in Gear 4:.457" diameter,.394" face width, 23 teeth/in Shaft Sizing ' Lspacer 2in Lthrust.5in Lend.25in Lcollar.25in Lgear23.25in Lbearing.5in Lgear45.5in Ltotal 2 Lbearing 2Lcollar Lspacer Lgear23 Lgear45 2Lthrust Lgear23 Ltotal L Lbearing Lthrust Lcollar 2 L Lgear45 Lgear23 L2 L Lspacer L2 3.75in in 5.25in Wt lbf Wt lbf Ltotal 5.25

21 Wr2 Wt2 tan ( ) Wr lbf Wr34 Wt34 tan ( ) Wr lbf Now the reactions will be determined at the end of the shaft. L.375 Wt2 Res2 Res lbf L cos ( ) Wt34 Res34 Res lbf cos ( ) Wt lbf Wt lbf Wr lbf Wr lbf Ray Rby Wr2 Wr34 Raz Rbz Wt2 Wt34 Rby Ltotal Rbz Ltotal Wr2( L) Wt34( L2) Wt2 L Wt34 L2 A Ltotal Ltotal B Wr2 Wr34 Wt2 Wr2 L Wt34 Wr34 L2 Wt2 L Wt34 L2 soln lsolve( AB ) soln lbf Ray 23.89lbf Raz lbf Rby lbf Rbz 253.7lbf Torque in shaft between gears: Tm Wt2 d lbf in For max bending moment.

22 Max moments; 4.5 lbf-in and -28 lbf-in from moment. Ma Mm 0 Ta 0 Assume 22 HR steel. a 4.4 b.78 ka asut b ka Assume diameter d.4 kb d.3 in.07 Sut 6.5 kpsi kc kd Fromt Table 6-5, 99% reliability za ke.08za ke 0.84 kf Se ka kbkc kdke kfsut Se 36.3 kpsi Using the DE-John Goodman Principle: Since the shaft is constant diameter, there will be no stress concentration; Kfs Kf n.5 Assume factor of safety of.5

23 D 6 n 2KfMalbfin Se0 3 psi 3( Kfs Tm) 2 2 Sut0 3 psi 3 D 0.358in Reference Table A-7 in Shigley, the preferred size is.4in D.4in Congruous with that required by the bore of the gear. Tm D 4.592lbf in 0.4in Computing the maximum stress on the shaft; max 2 32 Kf( Mm Malbfin) 6 Kfs Tm Ta D 3 3 ( ) D max psi f 2.094in f2.57in From Table 7-6, Key sizes: 3 w 32 in h 3 32 in Choosing 020 CD steel Sy 57 kpsi Tm F D lbf Force on the key at the surface

24 Now to check for failure by crushing: Assuming a factor of safety of 2 for the keys n 2 L 2F n wsy0 3 psi L 0.429in This is ok as the face of the gear is.4" and the gear has a hub. Because both gears have the same bore diameter and transmit the same load, the same key can be used for both. This concludes the mathematical analysis for the gear box.

Sheet 1 Variable loading

Sheet 1 Variable loading 1. Estimate S e for the following materials: a. AISI 1020 CD steel. b. AISI 1080 HR steel. c. 2024 T3 aluminum. d. AISI 4340 steel heat-treated to a tensile strength of 1700 MPa.