Stress Analysis of a Chicago Electric 4½ Angle Grinder

Transcription

1 Stress Analysis of a Chicago Electric 4½ Angle Grinder By Khalid Zouhri A Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute In Partial Fulfillment of the Requirements for the degree of MASTER OF MECHANICAL ENGINEERING Approved: Dr.Ernesto Gutierrez-Miravete, Project Adviser Rensselaer Polytechnic Institute Hartford, CT March, 010 (For Graduation August 010) 1

2 Copyright 010 by Khalid Zouhri All Rights Reserved

3 Table of Contents Abstract Introduction Data Force Analysis Stress Analysis Finite Element Analysis Conclusion References

4 List of Figures Figure 1: Chicago Electric Angle Grinder...7 Figure : Spiral bevel gear (large) and a spiral bevel pinion gear (small) Figure 3: Complete Component Assembly...8 Figure 4: Shaft with Bevel Pinion Gear...9 Figure 5: Shaft with Bevel Pinion Gear Removed...9 Figure 6: Spiral bevel pinion gear...10 Figure 7: SolidWorks Model Assembly...10 Figure 8: SolidWorks Assembly Exploded...11 Figure 9: Exploded view of components analyzed...11 Figure 10: ME- Rockwell Hardness Testing System...13 Figure 11: Pinion Close-up with Reference Lines...15 Figure 1: Schematic Drawing...16 Figure 13: Free Body Diagram of Shaft...16 Figure 14: Pinion- Bevel Schematic...17 Figure 15: Force Analysis -D Planes...19 Figure 16: Load, Shear Force, and Bending Moment Diagrams...1 Figure 17: Restraints at Bearing D Figure 18: Restraints at Bearing C...31 Figure 19: Shaft with Constraints and Loads and Study Run...3 Figure 0: Probe Value of Maximum Stress...3 List of Tables Table 1: Manufacturing Specifications...1 Table : Hardness Testing Results...13 Table 3: Hardness Conversion Chart Table 4: Data of the gear and Shaft

5 List of Symbols Symbol Definition b Tooth width (mm) or (in) d avg d out F S F a F r F t K v n p N g N p J P T V av K m K o K v Avg. diameter of pinion (mm) or (in) Outside diameter of pinion (mm) or (in) Factor of Safety Axial (thrust-bearing) load (N) or lbf Radial (separating) load (N) or (lbf) Tangential (torque-producing) load (N) or (lbf) Velocity Factor Rotating speed of pinion (rpm) Number of teeth on ring (driven) gear Number of teeth on pinion (driver) gear Geometry factor for bending strength Diametral pitch: large end of tooth (mm) Torque Avg. velocity (m/s.) Load distribution factor Overload factor Velocity Factor Φ Pressure angle ( ) Φ n Pressure angle in the plane normal to the tooth ( ) ψ Helix angle ( ) γ Pitch cone angle ( ) 5

6 Abstract The Electromechanical device Chicago 4.5 inch angle grinder used in many different applications. The device uses two gears, a bevel pinion gear and spiral pinion gear, the spiral bevel gear is attached to the shaft and to two other bearings. The components are driven by an electrical motor. The purpose of this project is to investigate the state of stress in the bevel pinion gear and in the shaft using standard hand calculations and finite element analysis (FEA) with the COSMOS software. 6

7 1. Introduction The purpose of this Engineering Project is to perform stress analysis of the electromechanical device Chicago 4.5 inch angle grinder ITEM 913-1VGA [Fig 1]. The grinder utilizes two spiral gears, a bevel gear and spiral bevel gear [Fig ] with output angular velocity (ω) 11,000 rpm. The pinion gear is directly coupled to the rotor shaft of the electric motor and supplies torque to the other bevel gear which is closed coupled to the output shaft. The pinion gear is held onto the end of the rotor shaft by a nut which forces the pinion against one of the two shaft bearings.the pinion gear will be the first component to be analyzed, the second component will be the rotor shaft of the electric motor. Figure 1: Chicago Electric Angle Grinder 7

8 Figure : Spiral bevel gear (large) and a spiral bevel pinion gear (small) Figure 3 shows the complete component assembly of the grinder. The motor is on the left, surrounding the rotor shaft. The gearing system is enclosed inside the gear box on the right. Figure 3: Complete Component Assembly 8

9 Figure 4 shows the spiral bevel pinion gear attached to the end rotor shaft by a nut. Figure 4: Shaft with Bevel Pinion Gear Figure 5 shows the two bearings attached to the end of the shaft (while the pinion gear has been removed for hardness testing Fig 6). Figure 5: Shaft with Bevel Pinion Gear Removed 9

10 Figure 6: Spiral bevel pinion gear Figure 7 shows the Solid Works representation of the assembly (bevel pinion gear, pinion shaft, bevel shaft, bearings, bevel gear, bevel casting, chuck washer, washer), Figure 8 shows an exploded view and Figure 9 shows a detailed view of the two components that will be analyzed, the bevel pinion gear and the shaft. Figure 7: SolidWorks Model Assembly 10

11 Figure 8: SolidWorks Assembly Exploded Figure 9: Exploded view of components analyzed 11

12 . Data Table 1 shows the given manufacturing specifications of the machine. Table 1 Manufacturing Specifications Angular Velocity 11,000 rev/min Voltage 110 Volts Frequency (AC) 60 Hz single phase Power 570 watts Amperage 4.5 amps For single phase armature motors one hp is generated by 70 watts. Assuming that the most efficiency expected out of a single phase motor is approximately 60% the motor power is calculated as 570w 3 hp hp 70w 4 For proper analysis, additional data must be collected. The material used in the angle grinder was unknown, therefore it was necessary to perform hardness testing to obtain material data. A Rockwell Hardness Testing Machine was used to measure the hardness of the shaft and bevel pinion gear. The bevel gear was completely isolated from its main subassembly. This ensured no cutting of material or extended surface hardening from clamping and pulling. The shaft was permanently fixed to its immediate subassembly. Luckily the shaft was just long enough to get true readings. Figure 10 shows the hardness test machine use d in this work. Before the hardness test, proper surface preparation was needed to ensure proper results. Three readings were done on the shaft as well as on the bevel in different locations to obtain an average hardness number. The location of testing on the material was also changed to get a variety of readings and to avoid tainted results from nearby surface hardening created by testing. It is imperative to note that the material will be much harder on the tooth of the gear rather than in the makeup material. There was no way to test the hardness of the material directly on the tooth without risking safety for the holder. The reading, however, was done as close as possible to the side of the tooth without risking the ball sliding down the face of the tooth. The results are shown in table. 1

13 Figure 10: ME- Rockwell Hardness Testing System Table : Hardness Testing Result Using a hardness conversion chart [] and the information from Shigley s Mechanical Engineering Design in Table 5 [3], material grade and tensile strength can be determined and the result for the shaft is shown highlighted in table 3. 13

14 Table 3: Hardness Conversion Chart For the gear, the following AGMA formula was use to calculate the tensile strength : S t 77.3H b + 1, 800 psi S tpiniom 77.3 * , 800 psi S tpiniom kpsi Additional geometrical data were also required for the analysis.the following calculations are based on the presentation given in Example 13.8 from Shigley s Mechanical Engineering Design (8th edition). As shown in figure 11 the outside diameter of the bevel pinion gear is d0.0mm, therefore the diametrical pitch is given by N 13 P t teeth / mm 0.65teeth / mm d 0 where N 13 Teeth 14

15 The helical angle needed to be determined. In the illustration below, lines were added and a protractor was used to measure it. Figure 11: Pinion Close-up with Reference Lines By using the reference lines the helical angle was measured to be 40 degrees. The normal diametrical pitch is calculated with the following formula P n P t cosϕ P n teeth / mm teeth / o cos 40 mm and the formula below is used to calculate the Pitch diameter N 13teeth d 15. mm p P teeth / mm 3 n The obtained data are summarized in table 4. Table 4: Data of the gear and shaft 15

16 3. Force Analysis In this section I perform force analysis using the methods described in Example 13:10 from Shigley s Mechanical Engineering Design 8 th edition Figure 1 shows the schematic drawing for the shaft with the bevel pinion gear. Figure 1: Schematic Drawing Figure 13 show the free body diagram for all the forces that apply to the shaft and bevel pinion gear and bevel gear. For each force we have tangential and axial and radial force Figure 13: Free Body Diagram of System 16

17 Gear Force Calculation Figure 14 show the pinion bevel schematic.when gears are used to transmit motion between intersecting shafts, some form of bevel gear is required. A bevel gearset is shown in Fig.14.Although bevel gears are usually made for a shaft angle of 90, they may be produced for almost any angle. The teeth may be cast, milled, or generated. Only the generated teeth may be classed as accurate. The pitch of bevel gears is measured at the large end of the tooth, and both the circular pitch and the pitch diameter are calculated in the same manner as for spur gears. It should be noted that the clearance is uniform. The pitch angles are defined by the pitch cones meeting at the apex, as shown in the figure. They are related to the tooth numbers as follows: tan y N/Ng tan T Ng/N Where the subscripts P and G refer to the pinion and gear, respectively, and where y and T are, respectively, the pitch angles of the pinion and gear. Figure 14 also shows that the shape of the teeth, when projected on the back cone, is the same as in a spur gear having a radius equal to the back-cone distance rb. This is called Tredgold's approximation. The number of teeth in this imaginary gear is N' rb/p Where N' is the virtual number of teeth and p is the circular pitch measured at the large end of the teeth. Standard straight-tooth bevel gears are cut by using a 0 pressure angle, unequal addenda and dedenda, and full-depth teeth. This increases the contact ratio, avoids undercut, and increases the strength of the pinion. Figure 14: Pinion- Bevel Schematic 17

18 To get the angle T we have to use formula below to get the y angle and from there we get T. PinionTeeth γ tan 1 ( ) Bevelteeth o.1 Γ o o o We use the formula below to get the distance t from the end of the bevel pinion gear to the pitch point.same way to get the y axis distance from the end of the bevel pinion gear to the pitch point. o t 3.5mm sin(.1 ) 1. mm 3 o t 3.5mm cos(.1 ) mm These thicknesses represent the location of the pitch point and are illustrated above in Figure 14. The distance from bearing D to the horizontal pitch point location P is: 9.5 mm is from outside of bevel pinion to far side of bearing Bearing is 7 mm wide so half of bearing is 3.50 mm DP( x) 19.5 (3.011mm mm) 13mm The distance from bearing D to the horizontal pitch point location P is: 0mm DP( y) 1.mm 8. 78mm d p 8.78mm 17.56mm( pitch diameter) We use the formula below to calculate the axial and Tangential and radial force that apply from the bevel gear into the bevel pinion gear. W t BP 3 60,000 hp 60,000 hp W t BP N π * d * n π *17.56mm *11,000rpm r W BP W t BP tan( Φ) cos( γ ) a W BP W t BP tan( Φ) sin( γ ) o W r BP 74.N tan 0 cos N o o W a BP 74.N tan 0 sin N o 18

19 v W { 10.16i j 74. k N W ABS( W ) N Shaft Force Calculations: Figure 15 shows the free body diagram for the shaft, with all the forces applied by the bearings and the bevel gear. Taking the moment around D: Figure 15: Force Analysais -D Planes (x-y) plane: So we take the moment at the point D to get the one equation with one unknown, and from there we solve for the force y F c M D F y c 136.5mm N 8.78mm 5.0N.13mm 0 F y c 1.73N ( change direction ) 1. 73N The total force in X direction equal to zero (static condition), to calculate the force x F D F x W a F X D 0 F x D N 19

20 The total force in y direction is equal to zero (static condition), to calculate the force F.. y D F y W r F y D + F y c 0 F y D N (x-z) Plane: Now we take the moment at the point D to get one equation with one unknown and from there we solve for the force z F c M D W t z c 13 mm + F.136.5mm 0 F z c 7.07N( change direction ) 7. 07N The total force in the z direction equal to zero (static condition), so the force z F D is then F z W t + F Z D F Z C 0 F Z D 81. 7N F c ( 1.73 j 7.07K N ) FC ABS( FC ) 7. 8N F D { 10.16i 6.75 j K} N F ABS( F ) 86. N D D Figure 16 shows the calculated and shear force and bending moment diagram. 0

21 Figure 16 : Load, Shear Force, and Bending Moment Diagrams 1

22 4. Stress Analysis Bending Stress and Bending Factor of Safety We use the formula below to calculate the Dynamic Factor (Kv). A V K v ( ) A The dynamic factor K v accounts for internally generated gear tooth loads which are induced by non-uniform meshing action (transmission error) of gear teeth. A quality number value of 6 is assumed. The transmission accuracy level number (Qv) could be taken as the quality number Qv 6. The American gear manufacturing association has defined a set of quality control numbers which can be taken as equal to the transmission accuracy level number Qv, to quantify these parameters (AGMA ). Classes 3< Qv<7 include must commercial quality gears and are generally suitable for any applications. B A V K v ( ) A B A (1 B) ifb 0.5(1 Qv ) 3 3 B 0.5*(1 6) A ( ) We use the formula below to calculate the velocity in m sec V π * d p * n p 60 * 17.56mm *11,000rev / min 10,113.8 mm sec m sec V π 60 K v ( m / sec )

23 The overload Factor (Ko), assuming uniform loading is Ko 1. The Size Factor Ks is calculated using table 14- [5] in Shigley s Mechanical Engineering Design. 8th Ed. (s, Yp (13T)and value is The width 6.5 mm 6.5 mm/5.4 mm/in in hence P Teeth/mm 1.55 Teeth/inch K s 1 k b F 1.19 ( Y P ) This is for standard units K s k b F Y 1.19 ( ) P 0.559in * *( ) 1.55T / in K H Cmf 1 + Cmc *( C pf C pm + Cma* C * e ) Assuming uncrowned so (load correction factor) Cmc1 F In (Pinion Proportion factor) Cpf; C pf F 10* d 6.5mm *17.56mm Cpm (pinion proportion modifier) 1.0 Cma(Mesh alignment factor) C ma A + BF + CF Using the table 14.9 [6], Value for A, B, C are as follows: A0.17 B

24 C C ma * Ce(Mesh alignment correction factor )1 K H C mf 1 + 1*(0.010* *1) The rim thickness factor Kb1 due to consistent thickness of gear. Speed Ration mg; m G N N G P 3 T.46 13T Load Cycle factor (Yn) using 13T for pinion and life cycle Y N ( P) * N p *(10 ) From Table [7] and a reliability of 0.90, KR (YZ) 0.85; From Figure 14-6 [8], The YJ(P) 0.1 KT (Temperature Factor) Temperature is less than 50 F so KT 1 St (Allowable Bending Stress for Hardened Steels) for 94 Brinell. Grade 1: St ( 0.533H B ) MPa (0.533* 94*88.3) 45MPa Sc (Contact fatigue strength) for Brinell and grade 1. Sc (.H B + 00) Mpa (.* ) 85. 7Mpa The pinion tooth bending stress is t P σ p W K 0K vks * * F K H K Yj B 1.55T / in 1.143*1 σ p 16.7ibf *1*1.59*.907 * * psi 76. 1Mpa 0.559in.1 4

25 We use the formula below to calculate the bending fatigue factor of safety for the pinion S F ( P) σ p St * YN K K T R S 45MPA* *1*.85 Mpa F ( P ) 3.7 σ p 76. 1MPa S F ( P) 3. 7 Wear Stress and Bending Factor of Safety Surface Condition Factor CfZr1(No surface defects specified) We use the formula below to show the stress and bending factor of safety Z Z p bp 1 [( r + a) r ] + [( r + a) r ] G bg 1 ( r p + r g )sinφ t Where; r p 8. 78mm r G mm r, bp r cosφt, rbg 16. bg 93 mm r bp 7. 93mm a 1 p n tan tan 0 tan 1 φn φt 5.41 o cosψ cos 40 Z 4. 71mm φ n o 0 o π π o pn pn cos Φ n cosφn cos mm p mm n o 5

26 m N p N 0.95Z 3.479mm.95* 4.71mm.778 I Z 1 cos 0 sin 0 * m o N o m * m G G + 1 I Z cosφ *sinφ.46 * * From the table 14-8 [9] from Shigley s we calculate the value of Ze (Elastic coefficients factor); C p Z E 191 MPa 300 psi We use the formula Stress cycle factors for pinion; ( ) *(10 ). 948 Z N ( P) * N p Finally we use the formula below to calculate the contact stress: C t K H σ c C p W K 0Kv K s * d * F I ( p) f σ psi 16.7ibf *1*1.59*.907 * * 74.7ksi 515MPa.69134in * and use the formula below to calculate the wear Factor of safety: S sc * z [ KT * K σ N ] 85.7MAP *.948 [ ] 1* MPa R H ( P) C 1.85 σ 515MPa S H ( P) Stress Analysis of Output Shaft using the vector cross product We use the formula below to get the absolute value for W 6

27 W { 10.16i j 74.k} N W ABS( W ) N R DP { 13i 8.78 j} mm R DC {136.5 i} mm From both equations above we use the values to calculate the torque T t Bp. T 8.78mm * W 8.78mm *74.N 651N mm T 651.5iN. mm Taking the moment around D to calculate the force C F Z : M D R DP * W + R * F + T 0 DC C M D { 13i 8.78 j} mm *{10.16 i j 74.k} N + {136.4 i} mm *{ F j + F k} N + { Ti} 0 c y c z N mm F C Z 7. 07KN 136.5mm 36.1N. mm Y 1.73 j.. N 136.5mm F C F C { 1.73 j 7.07k} N Fc ABS ( FC ) 7. 8N To get Fd, we use the equation below (sum the forces equal to zero, static condition). 7

28 F d + F + W C 0 Since the sum of the force equal to zero (static condition) F d x D y D Z { F i + F + F } lbf + {0i j 7.07} N + {10.16 i j 74.} N 0 Fx d i N F d y j. N 81.3K N Fz d. F { 10.16i 6.75 j k} N The absolute value for Fd is calculated below. FD ABS( F D ) 86. N The safety factor of the shaft. We use the formula below to calculate the moment on the shaft at point D of the bearing. max M ( 965N. mm) + (36.N. mm) 993.5N. mm M max 993.5N. mm T max 651.5N. mm D / d 10.10mm / 8mm 1.6 Assume r.54 mm (0.1in) r / d.54mm /8mm.3175 K t 1.4[1] K ts 1.[13] 8

29 Notch sensitivity (q) 0.95[14] Shear notch sensitivity (qshear)1 [15] K f 1 + q( Kt 1) (1.4 1) 1.38 K 1 + q ( K 1) 1+ 1(1. 1) 1. fs s ts " 3. M max 3 *993.5N. mm σ a K f MPa 3 3 π. d π.(8mm) " m σ 3.[ K fs 16. T π. d max 3 ] " 16.*651.5N. mm σ m 3.[1. ] MPa 3 π.(8mm) Assume Sy0.775Sut Sut106Ksi 730.8MPa So the safety factor of the shaft is calculated using the formula below. n yield ' m S y σ + σ ' a n yield 0.775* 730.8MPa 13.47MPa + 7.8MPa 13.9 Fatigue Factor of Safety Calculations Analysis based on Modified Goodman Design Theory ' e 0.5 S ut S * ' S e 0.5* 730.8MPa MPa Using the table 6-[5] from Shigley s and assuming a surface finishes of mechanized or cold drawn: a4.51 b

30 Surface Factor Ka; K a a * S b ut 4.51* 730.8MPa Combined Loading so Kc1 Room Temperature operation so kd1 From table 6-5[4] from Shigley s and also assuming 90% reliability ke0.897 ' Se K a KbKc K d KeSe.786*.993*1*1*.897 * 365.4MPa 55. 8MPa n 1 fatigue σ S ' a e σ + S ' m ut n fatigue S ut S e ' a * S ut * σ + S. σ e ' m n fatigue 55.8MPa *730.8MPa 730.8Mpa * 7.8MPa MPa MPa 8 Comparing N yield and N fatigue n yield 13.9 > n 8 fatigue This shows that the shaft will fail by fatigue. 30

31 5. Finite Element Analysis Figures 17 and 18 show the load and restraint setup on Cosmos software. The green arrows in the pictures represented the mechanical constrain impose by bearings D and C on the shaft. At bearing D, radial and axial restraints were set. Restraints were applied to a mm cylindrical face placed at the midpoint of the bearing location. The bearing restraint at C was set for only a rotational restraint. This was to ensure that over-constraining the shaft was avoided. This restraint was also applied to a mm cylindrical face placed at the midpoint of the bearing location. Figure 17: Restraints at Bearing D Figure 3: Restraints at Bearing C Figure 18: Loads at Pitch Point Applied Using Remote Load 31

32 Figure 19 shows the shaft with constraints and loads and the computed values of the von Mises stress are also shown. The maximum stress (8.1 MPa ) appears on the bearing area.an expanded view of the most highly stressed area is shown in figure 0. Figure 19: Shaft with Constraints and Loads and Study Run ' σ max Figure 0: Probe Value of Maximum Stress 3

33 Using Von Misses to get an analytical value for ' σ max ' max σ m σ m * σ a + σ a 3. τ m σ + σ 3*965N. mm (1.38* ) π *(8mm) 16*36.N. mm + 3.(1. ) 3 π.(8mm) ' max 6.9MPa % Difference : x1 x % Difference *100 x1 + x 6.9MPa 8.1MPa % Difference * % diffrence 6.9MPa + 8.1MPa Using COSMOS, the maximum value found was 8.1 MPa. There is a difference of 4.43 % from a calculated value of 6.9MPa. This could be attributed to model building techniques in SolidWorks. There could also be variations in part measurements which could skew calculations slightly. Changes in restraint locations and bearing restraints types may bring different results. Better understanding and practice of COSMOS could also produce improved results. Overall the stress analysis attributed by COSMOS is an excellent representation of the type of stress the part may succumb to. 33

34 6. Conclusion The project exemplified real world tasks that definitely sharpened my skills a young engineer. The hands-on experience gained by using the laboratory equipment and software tools was invaluable. The experience of using specific tools, acquiring data, and using this data in calculations was a particularly useful experience. It provided some relevance of the knowledge learned from work experience and my other engineering courses. Also COSMOS in particular was beneficial not only because I was able to use it as a comparison tool but also because of the practical experience gained which will be useful in the a real-world application. This tool gives a visual representation of the types of stresses and possible deformations that may occur in design. This quick reference will give the design engineer an approximation of what to expect under certain loads or applications. This brief exposure to COSMOS will certainly be invaluable. Reference [1] Manufacturing Specifications link, p/english/gear/myset/spiral.html [] Hardness conversion chart, [3] AGMA Strength Graph for Hardened Steels, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. pp 77 [4] Reliability Assumption, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Table 6-5, pp 85 [5] Parameters for Marin Surface Modification Factors, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Table 6-, pp 80 [6] Lewis Form Factor, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Table 14-, pp

35 [7] Empirical Constants for Load Distribution Factor Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Table 14-9, pp 740 [8] Reliability Factor KR Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Table 14-10, pp 744 [9] Geometry Factor, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Figure 14-6, pp 77 [10] For Elastic Coefficient, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Table 14-8, pp 737 [1] For Stress Concentration Value Kt, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Appendix Figure 15-6, pp 1007 [13] For Stress Concentration Value Kts. Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Appendix Figure 15-8, pp [14] Notch Sensitivity q, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Figure 6-0, pp 87 [15] Shear Notch Sensitivity q, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Figure 6-1, pp 88 [16] pitch diameter p, Budynas, R., Nisbett, K., Shigley s Mechanical Engineering Design. 8th Ed. McGraw Hill, New York NY, 008. Example 13:15, pp 45 35