### Face Width, F

• “Face width is the width of the gear parallel to the axis of the gear. It is defined by the designer as one of the required design decisions. More is said about face width in Chapter 9, where strength of the teeth is considered. For now, we can state that a nominal value for face width is approximately F ~ 12 / Pd, but a wide range is permitted.” (Machine Elements in Mechanical Design, 5th by Robert L. Mott, p. 282)

### Diametral Pitch

• Barber-Colman Company, Loves Park, IL, referenced from Machine Elements in Mechanical Design, 5th by Robert L. Mott, p. 279.

#### Spur Gears - Drafting and Design

##### Part 1 - Determine the four Gear Circle Diameters (Pitch, Outside/Addendum, Root, and Base)
• Step 1 - select size of gear teeth (Diametral Pitch, P)
• “Diametral Pitch (D.P.). This is certainly by far the most common and the most useful method of notation for small gears and the definition of a small gear in this case is any gear that the model engineer or backyard amateur is likely to handle. The diametral pitch is simply the number of teeth a wheel has per inch of pitch diameter. For example, if a gear has a P.C.D. (Pitch Circle Diameter) of 2in. diameter and has 40 teeth then it is said to be 20 D.P. If the number where to be 40 on a P.C.D. of 1 in. diameter then the D.P. would be 40. D.P.s are usually whole numbers and, more often than not, even numbers. It is not usual to encounter an odd D.P. number after 10 D.P. has been reached and gears below 10 D.P. will in the main be larger than the amateur will want to cut. This arrangement makes the setting out of gear train centres easy. For example, should it be decided to product two gears to give a ratio of 3:1 and 20 D.P. was chosen for the tooth size, two gears - one with 20 teeth and one with 60 teeth - would appear to be satisfactory. The P.C.D. of the 20 tooth would be 1 in. whicle the P.C.D. of the 60 tooth would be 3 in. This would mean that the gear centre would be 2in., or half of the addition of the two P.C.Ds. (Gears and Gear Cutting by Law, p.30)
• “Diametral pitch is not a pitch in the same sense as the preceding pitches. It represents the size of the tooth. The larger the numeric value of the diametral pitch, the smaller the size of the gear tooth… Many fine-pitch gears are produced by means of generating tooling. Even gears produced by molding, casting or stamping are intended to have teeth which are the same as if they were generated. Gear generating tools, such as hobs, shaper cutters and racks, have a basic tooth form which, when cutting a gear on a generating machine, produces involute gear teeth. Although a hob, for example, can generate gears having any desired number of teeth, it can only produce a single normal diametral pitch and normal profile angle. Since several tools may be needed to produce a job lot of gears, it is generally desirable to select a diametral pitch for which most gear shops are likely to have tooling. This may avoid the need to purchase special tooling. Thus the following have become recommended normal diametral pitches, Pnd = 20, 24, 32, 40, 48, 64, 72, 80, 96, 120 (AGMA 917-B97 Design Manual for Parallel Shaft Fine-Pitch Gearing, p. 16) (secure PDF)
• Dudley's Handbook of Practical Gear Design and Manufacture, 2nd (secure PDF) gives the following Diametral Pitch, D recommendations
• “Pitch Diameters Obtained with Diametral Pitch System.-The diametral pitch system is arranged to provide a series of standard tooth sizes, the principle being similar to the standardization of screw thread pitches. Inasmuch as there must be a whole number of teeth on each gear, the increase in pitch diameter per tooth varies according to the pitch. For example, the pitch diameter of a gear having, say, 20 teeth of 4 diametral pitch, will be 5 inches; 21 teeth, 5 1/4 inches; and so on, the increase in diameter for each additional tooth being equal to 1/4 inch for 4 diametral pitch. Similarly, for 2 diametral pitch the variations for successive numbers of teeth would equal 1/2 inch, and for 10 diametral pitch the variations would equal 1/10 inch, etc.” (Machinery's Handbook, 27th Edition by Industrial Press, p. 2034-5 secure PDF)
• select a Diametral Pitch, P. Must be a whole number like 1, 2, 3, 4,….
• “Since diametral pitch is used only with U.S. units, it is expressed as teeth per inch.” (Shigley's Mechanical Engineering Design, 8th, p. 656)
• Figure showing comparative sizes and shape of gear teeth. The higher the P the smaller the teeth
• “Spur gear design normally begins with selecting pitch diameters to suit the required speed ratio, center distance, and space limitations. The size of the teeth (the diametral pitch) depends on the gear speeds, gear materials, horsepower to be transmitted, and the selected tooth form.” (Technical Drawing with Engineering Graphics, 14th Ed by Giesecke, Mitchell, Spencer, Hill, Dygdon, Novak, & Lockhart, p. 653)
• “Diametral pitch is the ratio of the number of teeth to the number of inches in the pitch diameter in the plane of rotation, or the number of gear teeth to each inch of pitch diameter. Normal diametral pitch is the diametral pitch as calculated in the normal plane, or the diametral pitch divided by the cosine of the helix angle.” (Machinery's Handbook, 27th Edition by Industrial Press, p. 2030 secure PDF)
• “Diametral and Circular Pitch Systems.-Gear tooth system standards are established by specifying the tooth proportions of the basic rack. The diametral pitch system is applied to most of the gearing produced in the United States. If gear teeth are larger than about one diametral pitch, it is common practice to use the circular pitch system. The circular pitch system is also applied to cast gearing and it is commonly used in connection with the design and manufacture of worm gearing.” (Machinery's Handbook, 27th Edition by Industrial Press, p. 2034 secure PDF)
• Step 2 - select Number of Teeth, N
• Number of teeth must be a multiple of diametral pitch, P
• Number of teeth of large gear/wheel, NG. Say P = 4, then NG must be a multiple of 4 (i.e. 4, 8, 12, 16, 20,…). Lets use NG = 44.
• Number of teeth of small gear/pinion, NP. Also, NP must be a multiple of 4. Lets use NP = 24
• “The minimum number of teeth Nmin of standard proportion that may be cut without undercut is:
• Nmin = 2P csc2φ[aH- rt(1 - sinφ)]
• where aH = cutter addendum; rt = radius at cutter tip or corners
• φ = cutter pressure angle; and P = diametral pitch.
• (Machinery's Handbook, 27th Edition by Industrial Press, p. 2058 secure PDF)
• Step 3 - calculate gear ratio, mG
• “The teeth on mating gears must be of equal width and spacing, so the number of teeth on each gear, N, is directly proportional to its pitch diameter, or mG = NG / NP” (Technical Drawing with Engineering Graphics, 14th Ed by Giesecke, Mitchell, Spencer, Hill, Dygdon, Novak, & Lockhart, p. 648)
• Example mG = 44/24 = 11/6 or 11:6 (read as 11 to 6)
• Step 4 - select a common Pressure Angle, f (14.5°, 20° or 25°) between the larger gear/wheel and the small gear/pinion
• “The angle f is called the pressure angle, and it usually has values of 20° or 25°, though 14.5° was once used.” (Shigley's Mechanical Engineering Design, 8th, p. 659)
• “The involute tooth form depends on the pressure angle, which was ordinarily 14.5° and is now typically 20° or 25°. This pressure angle determines the size of the base circle; from this the involute curve is generated.” (Technical Drawing with Engineering Graphics, 14th Ed by Giesecke, Mitchell, Spencer, Hill, Dygdon, Novak, & Lockhart, p. 650)
• “The dimensions relating to tooth height are for full-depth 14.5° (which are becoming outmoded) or with 20° or 25° involute teeth. Of course, meshing gears must have the same pressure angle.” (Technical Drawing with Engineering Graphics, 14th Ed by Giesecke, Mitchell, Spencer, Hill, Dygdon, Novak, & Lockhart, p. 648)
• Step 5 - calculate Pitch Diameter (D, DG, DP) (circle 1 of 4)
• Is the diameter of the pitch circle of the gear or pinion
• Pitch Diameter, D = Number of Teeth, N / Diametral Pitch, P
• DG = NG / P = 44/4 = 11”
• DP = NP / P = 24/4 = 6”
• Center Distance, C = (DG + DP)/2 = R + r = (11” + 6“)/2 = 8.5”
• Step 6 - calculate outside diameter, DO (Addendum Circle) (circle 2 of 4)
• DO = D + 2a where a = Addendum
• Diameter of addendum circle, equal to pitch diameter plus twice the addendum.
• a = 1/P, example a = 1/4
• DOG = DG + 2a = 11“ + 2(1/4)” = 11.5“ for the larger gear/wheel
• DOP = DP + 2a = 6” + 2(1/4)“ = 6.5” for the smaller gear/pinion
• “Outside diameter is the diameter of the circle that contains the tops of the teeth of external gears.” (Machinery's Handbook, 27th Edition by Industrial Press, p. 2031 secure PDF)
• Step 7 - calculate the Root Diameter, DR (circle 3 of 4)
• DR = D - 2b where b = Dedendum
• b = 1.25/P when f = 20° or 25°, example b = 1.25/4 = 0.3125
• b = 1.157/P when f = 14.5°
• DRG = DG - 2b = 11“ - 2(0.3125”) = 10.375“
• DRP = DP - 2b = 6” - 2(0.3125“) = 5.375”
• Step 8 - calculate the base circle diameter, DB (circle 4 of 4)
• “The involute is generated from the base circle the diameter” (Handbook of Gear Design, 2nd by Gitin M. Maitra, Appendix A - Construction of Involute Gear Tooth)
• DB = D cos f
• DBG = DG cos 20° = 11“ cos 20° = 10.3366”
• DBP = DP cos 20° = 6“ cos 20° = 5.6382”
##### Part 2 - Drafting the four Gear Circles
• Step 9 - Drafting Gear Teeth
• “Involute gears are interchangeable when they have set conditions that allow them to mesh properly. The four conditions for interchanging involute gears are the following: the same diametral pitch, the same pressure angle, the same addendum, and the same dedendum.” (Mechanical Drawing, 10th Ed by French, Svensen, Helsel, & Urbanick, p. 407)
• Autodesk Community - Inventor General True involute gear generation
• Background
• “The gear ratio is also the pitch diameter of the gear divided by the pitch diameter of the pinion. For example, if the gear has a pitch diameter of 4 and the pinion has a pitch diameter of 1, then the gear ratio is 4:1, and the pinion ratio is 1:4.” (Technical Graphics Communication, 4th by Bertoline, Wiebe, Hartman, and Ross, p. 1112)
##### Approximate Involute Curve Method Spur Gears in Technical Drawing
• How to draw a spur gear tooth from Technical Drawing, 4th by Goetsch and Chalk, p. 596 (secure PDF)
• Step 0: Draw the following Spur Gear
• Step 1: Spur gear specifications
• Diametral Pitch (tooth size), P = 4
• Pitch Diameter, DP = 6.0“
• Number of Teeth, N = 24
• Pressure angle, f = 20°
• Step 2: Calculate Base Circle
• Base Circle Diameter, Db = DP cos f
• Db = 6.0” cos 20° = 5.638“ (Rb = 5.638” / 2 = 2.819“)
• Draw the Base Circle in Inventor (use color Blue)
• Step 3: Calculate approximate involute profile curve radius (Rapprox involute curve)
• Rapprox involute curve = Pitch Diameter / 8 = D / 8 = 6.0” / 8 = 0.75“
• Draw two circles with (Rapprox curve = 0.75”) anywhere outside the Base Circle (use color Green)
• Step 4: Apply coincident constraints between edge of Base Circle and center of green circle (approximate involute curve)
• apply a coincident constraint between the center of the 0.75“ radius circle and anywhere along the base circle
• Step 5: Draw Pitch Circle, DP = 6.0” (use color Pink)
• Step 6: Draw the radial lines for the circular pitch arc
• draw two lines from the center of the pitch circle to the edge of the pitch circle (pink circle)
• assign a dimension to the tooth width angle = 360° / (N * 2) = 360° / (24 * 2) = 7.5°
• Alternative - Calculate Circular Pitch, p distance (not chord distance)
• Circular Pitch, p = p / Diametral Pitch ? P = 3.14159 / 4 = 0.785“
• tooth thickness = p / 2 = 0.785” / 2 = 0.392“
• “Circular pitch is the distance on the circumference of the pitch circle, in the plane of rotation, between corresponding points of adjacent teeth. The length of the arc of the pitch circle between the centers or other corresponding points of adjacent teeth.” (Machinery's Handbook, 27th Edition, Industrial Press, p. 2030)
• Step 7: add two points at the intersection of the radial lines with the Pitch Circle (use black color)
• Step 8: apply another coincident constraint between these two points and the approximate involute circles with rapprox. involute = 0.75”
• Step 9: delete the radial lines (pink)
• Step 10: create two new radial lines (color blue) from the center of the base circle (blue circle) and the intersection with the approximate involute circles (green circles).
• Step 11: calculate the tooth dedendum, b height
• b = 1.250 / P = 1.250 / 4 = 0.3125“
• Step 12: calculate the root circle diameter
• DR = D - 2b = 6” - 2 * 0.3125“ = 5.375”
• Step 13: draw the root circle (color red)
• Step 14: trim the approximate involute circle (green circle) between the base circle (blue circle) and the root circle (red circle)
• Step 15: draw the outer circle (color orange)
• DO = (N + 2) / P = (24 + 2) / 4 = 6.5“
• Step 16: trim top of gear tooth
• delete outer circle (orange color)
• delete approximate involute circle (green circle)
• delete pitch circle (pink color)
• delete work points (black color)
• delete base circle (blue color)
• Step 17: draw radial lines (red color)
• Step 18: apply dimension, interior angle of 360° / Number of Teeth, N
• interior tooth angle = 360° / 24 = 15°
• Step 19: trim blue radial lines and root circle (red). Basically trim everything until left with a single tooth and blank sides
• Step 20: apply root fillet, rf
• rf = 0.3 / P = 0.3 / 4 = 0.075”
• Step 21: finish sketch
• Step 22: extrude tooth wedge say 1“
• “Face width-to-diameter ratio. To achieve more uniform tooth contact along the face, the ratio of face width to diameter should usually be held to below 2.0.” (AGMA 917-B97, p. 29)
• Face Width, F = 2 * Pitch Diameter, D
• “Usually the face width of one of the gears is made wider to allow for axial misalignment and still maintain full face contact. The increase in face width is usually made to the pinion because it requires less additional material.” (AGMA 917-B97, p. 29)
• Advantages of helical gears: “Generally quieter than spur gears if the active face width is greater than one axial pitch.” (AGMA 917-B97, p. 25)
• Step 24: use circular pattern for the number of teeth, N = 24
• Reference
• Technical Drawing, 4th Edition by Goesck (secure PDF)
##### Spur Gear and Inventor Design Accelerator
• Autodesk Inventor 2014 Help - Spur Gears Component Generator
• Module. Older text books quote the module as being the reciprocal of the D.P. (Diametral Pitch) but more recently it has become the 'metric' way of quoting the size of the teeth. The module can be said to be the pitch diameter in millimetres divided by the number of teeth, or to put it the other way round, the P.C.D. (Pitch Circle Diameter) in millimetres is the module number multiplied by the number of teeth in the gear. As there are 25.4 millimetres to one inch then a number 1 module is equal to 25.4 D.P., a number 2 module would be 12.7 D.P. whilst a .5 module would be 50.4 D.P.” Gears and Gear Cutting, Workshop Practice Series 17 by Ivan Law, 1990 by Argus Books (secure PDF)
• Create the spur gear (pinion and gear/wheel) in Figure 1-0 using the Inventor Design Accelerator.
• Formulas needed to calculate inputs for the Inventor Design Accelerator
• Step 0. Assume pressure angle, f = 20°
• Step 1. Calculate the Gear Ratio, mG
• mG = DG/DP = 5.25”/3.75“ = 1.4
• Step 2. Calculate the Center distance, C
• C = (DG + DP)/2
• DG = 5.25”
• DP = 3.75“
• C = (5.25” + 3.75“)/2 = 4.5”
• Step 3. Given the Outside/Addendum Diameter, DO
• DOG = Outside Diameter of large gear/wheel/spur = 6“
• DOP = Outside Diameter of small pinion gear = 4.5”
• Step 4. Calculate Addendum, a
• DO = D + 2a ? a = ( DO - D ) / 2
• aG = ( DOG - DG ) / 2 = ( 6“ - 5.25” ) / 2 = 0.375“
• aP = ( DOP - DP ) / 2 = ( 4.5” - 3.75“ ) / 2 = 0.375”
• for gears to mesh, must have the same gear tooth profile, that is Addendum heights are equal, aG = aP
• Step 5. Calculate Diametral Pitch, P
• a = 1 / P ? P = 1 / a
• P = 1 / 0.375 = 2.67
• Given P = 21/32 = 0.65625
• so, why aren't these values the same?
• Step 6. Inventor Assembly ? Design ? Spur Gear
• Design guide: Module and Number of Teeth
• Pressure Angle: 20°
• Desired Gear Ratio: 1.4
• Center Distance: 4.5“
##### Spur Gear Calculations
• Background
• Angular coordinate is expressed in radians (rad) or occasionally in degrees (°) or revolutions (rev)
• 1 rev = 2p rad = 360° (Vector Mechanics for Engineers Statics and Dynamics 9th by Beer, p. 919)
• Example: DG = 27” and DP = 9“, what is the gear ratio, mG
• Answer: mG = DG / DP = 27” / 9“ = 3:1
• (see Technical Drawing with Engineering Graphics 14th Edition by Giesecke, p. 648)
• Discussion of angular momentum, see Conceptual Physics, 9th by Hewitt
• Example: DG = 27” and DP = 9“ and nP = ?P = 1725 rpm, what is the angular velocity of the large gear/wheel (nG)
• Answer: nG = nP (DP / DG) = 1725 rpm (9” / 27“) = 575 rpm
• (see Technical Drawing with Engineering Graphics 14th Edition by Giesecke, p. 648)
##### Gear Manufactures and Reference
• Manufactures
• WM Berg Robert Marszalkowski - Application Engineering, 414-747-5808, wmbergtechsupport@wmberg.com, 5138 S. International Dr, Cudahy WI 53110
• Winzeler Gear producer of stamped metal gears for radio, appliance and toy industries. email: jwinzelergear.com, phone: 708-867-7971, 355 W Wilson Ave, Harwood Heights, IL 60706
• A 1M 2-Y24056 Plastic Spur Gear Diametral Pitch, D=24 (24 and 32 are very common for SDP/SI)
• Shipping and handling cost was \$11.41 for just two plastic gears
• Moore Machine and Gear, Inc Alan Moore, email: moorecustomgear@gmail.com or mooregear@insightbb.com recommends Stock Drive Products / Sterling Instrument 800-645-1144 for low quantities of gears
• Omni Gear & Machine Corp email: info@omnigear.us, John Hall (jhall@omnigear.us)
• Precipart email: gearedsolutions@precipart.com <strike>info@precipart.com</strike>
• Rapid Gear email: dave.scapinello@rapidgear.com in Canada
• Quality Transmission Components email: qtcsupport@qtcgears.com
• United Gear and Assembly email: <strike>customerservice@ugaco.com</strike>
• Worcester Gears & Racks email: <strike>mail@wgear.com</strike>
• Youtube.com video Gear Train Calculations
• Gears can be animated using the Motion Constraint or the Contact Solver. The Motion Constraint just animates a rotation of the gear and requires the input of a correct gear ratio. The Contact Solver is a more realistic animation of the gears because the second gear will only move when there is contact with the first gear. In the case of the Motion Constraint, there is no contact and the gears just rotate.
• Step 1. Create Base Plate.ipt (5”x5“ square with extruded thickness of 3/8”=0.375“)
• Step 2. Create Bearing Housing.ipt
• Balloon Note 1 is for a chamfer of 0.062”
• Step 3. Add the above parts to an Inventor Assembly and fully constraint the parts
• Add two instances of Bearing Housing.ipt
• Apply a mate-mate constraint between the top face of Base Plate.ipt and the bottom face of Bearing Housing.ipt
• Base Plate should be grounded
• Bearing Housing 1 is mate constraint 1.5“ from the centerline of the housing to the edge of the base plate. Do this on two sides.
• Bearing Housing 2 is only mate constraint 1.5” from one edge of the base plate. Then apply a 3“ mate constraint between the two bearing housing.
• Assemble ? Productivity - Degree of Freedom Analysis to confirm the assembly is fully constrained
• Step 4. Add Bolted Connection
• Design ? Bolted Connection
• Design Tab: Type through all
• Design Tab: Placement - On Point
• Design Tab: Start Plane - select the top plane of the Bearing Housing (where the bolts will go)
• Design Tab: Termination - select the bottom of the Base Plate
• Design Tab: Thread - ANSI Unified Screw Threads - 0.164 #8 inch
• Design Tab: ANSI Oval Head Machine Screw applied to the Bearing Housing
• Design Tab: select to add a fastener, select ANSI Cross Recessed Oval Countersunk Head Machine Screw - Type I - inch
• Design Tab: Click to add a fastener, Regular helical Spring Lock Washer (Inch)
• Design Tab: Click to add a fastener - Hex machine screw nut - inch
• Design Tab: save as Templates Library by clicking the Add button, Bolted Connection name
• Step 5. Spur Gear Design Accelerator
• Gear References
• Liansuo Xie, Ph.D. is the Global CAD Manager (lian.xie@marel.com 515-263-3331) at Marel Meat Processing Inc and has written an Inventor add-in that creates the actual involute curve for the gear tooth.
• Differential Gears by A Jam Handy Productions
• Horsepower (1937) by A Jam Handy Productions
• A Jam Handy - Jamison Handy - pioneer in the field in video education.  