how to draw involute gear tooth profile

The Involute Curve, Drafting a Gear in CAD and Applications

Past Nick Carter

Introduction:

Well-nigh of us attain a point in our projects where we have to brand use of gears. Gears can be bought ready-fabricated, they can be milled using a special cutter and for those lucky enough to accept access to a gear hobber, hobbed to perfect form. Sometimes though we don't have the coin for the milling cutters or gears, or in search of a project for our own edification seek to produce gears without the aid of them. This commodity will explain how to draw an involute gear using a graphical method in your CAD program that involves very little math, and a few ways of applying it to the industry of gears in your workshop.

The method I describe volition allow yous to graphically generate a very close approximation of the involute, to any precision you desire, using a simple 2D CAD programme and very little math. I don't want to run though familiar territory so I would refer you to the Machinery's Handbook'southward affiliate on gears and gearing which contains all the bones information and nomenclature of the involute gear which you volition need for this exercise. Some higher terminate CAD programs already have functions for generating the involute tooth from input parameters, but where'southward the fun in that?

When we talk nearly gears, about of us are talking nearly the involute gear. An involute is all-time imagined by thinking of a spool of string. Necktie the end of the string to a pen, start with the pen against the border of the spool and unwind the cord, keeping it taut. The pen will depict the anfractuous of that circumvolve of the spool. Each molar of an involute gear has the profile of that curve equally generated from the base circle of the gear to the outside diameter of the gear.

Drawing a Gear:

The easiest style to teach is to demonstrate, and then here are our parameters for drawing a:

16 Diametral Pitch (P), 20 Molar (Due north), 14-one/2 Pressure Angle (PA) involute spur gear

• We need to compute the post-obit information; a scientific calculator is handy for figuring out the cosine of the pressure bending.
  ("/" Denotes segmentation, "*" denotes multiplication, the dedendum (d) is computed differently for other pressure angles; see Machinery'southward Handbook for the correct formula.)
  
• The Pitch Diameter (D) = N/P = 20/16 = i.25"
• The Pitch Radius (R) = D/2 = .625"
• The Base Circle Diameter (DB) = D * COS (PA) = ane.25 * COS (14.5 deg) = 1.210"
• The Base Circle Radius (RB) = DB/2 = .605"
• The Addendum (a) = ane/P = i/xvi = .0625
• The Dedendum (d) = 1.157/P = i.157/sixteen = .0723" (rounding off at .0001")
• Outside Diameter (Practise) = D+2*a = i.375"
• Outside Radius (RO) = .625" (R) + .0625" (a) = .6875"
• Root Diameter (DR) = D-2*d = 1.1054"                              (I had before used the letter of the alphabet "b" instead of "d", I recollect this was a typo, making a note here on 10/09/10 of the revision)
• Root Radius (RR) = .625" (R) - .0723" (d) = .5527"            (I had earlier used the letter of the alphabet "b" instead of "d", I think this was a typo, making a note here on x/09/10 of the revision)

For our method we need to compute the following also:

1. Circumference of the Base circle, (CB) = Pi * (DB) = Pi * 1.210" = 3.8013"
2. i/20th of the Base Circle Radius, (FCB) = .03025"
3. Number of times that FCB can be divided into CB, (NCB) = 125.6628
4. 360 degrees divided by NCB, (ACB) = two.86 degrees
5. Gear Tooth Spacing (GT) = 360/Due north = 18 degrees
6. The 1/20th of the Base Circle Radius (FCB) is an arbitrary division, which yields a very close approximation; you can use whatsoever fraction you think will yield a good result. Now we take all our pertinent information, let'due south get drawing!
   

Annotation: Click on any cartoon to download a .dxf file of that drawing.

Open upwards your CAD program and describe concentric circles of the Pitch Diameter (D), Base Circle diameter (DB), Exterior Diameter (Practise), and Root Diameter (DR). Add a circle of .25" diameter for the bore of the gear. Make sure the circle centers are at ten=0, y=0.
1) Draw a line from the circle center (0,0) to the base of operations circumvolve perpendicular to your grid. In other words at 0, 90, 180 or 270 degrees. I chose 270 degrees. 2) Draw a line 1/20th of the Base Circle Radius (RB) long (FCB = .03025") at a right angle from the end of that line. This line is now tangent to the base circle. It volition be very hard to see unless y'all zoom in on the intersection of the base circle and the lines. 3) Radially copy the two lines (eye at 0,0), make 14 copies at 2.86 degrees autonomously (ACB), for a total of fifteen line pairs. Depending on the bore of the gear you may demand more or less lines, smaller gears need more, larger gears may need a smaller fraction of RB (base circumvolve radius). four) Number each set of lines, starting with 0 for the offset 1, going to xiv Drawing shows the two lines, and the copies of the line laid out and numbered.
5) Extend the tangent line for each re-create then it's length is the 1/20th of the base circle radius (FCB) times the number that you have next to that tangent line (0 ten FCB, 1 x FCB, 2 10 FCB…xiv x FCB) extend them from the tangent point. Nigh CAD programs will make this very easy, providing that you started the line from tangent betoken, usually you just alter the length parameter for each line, in Autosketch at that place is a display showing the line data and retyping the length extends the line from its start bespeak. Make sure y'all zoom in on the drawing so you extend the correct line. Drawing shows the tangent lines extended, and the length of tangent #14, which is .4235" or FCB x xiv
six) Starting at tangent line #0, draw a line from the end of tangent #0 to so end of tangent #1, from the terminate of tangent #i to tangent #2, tangent #two to tangent #iii so on. Yous should now have a very close approximation of the involute bend starting at the base circle and extending past the addendum circle. Trim the involute curve to Practice, the outside diameter of the gear. Cartoon shows the involute drawn along the ends of the tangent lines.
7) Erase all the tangent lines, leaving the involute bend generated past the process. Make a line that goes from the intersection of the anfractuous bend and the pitch diameter circle (D) to the center of the gear. Annotation that this will not be the same as the line going from the first of the involute at the base circumvolve (DB) to the center. 8) Draw a second line � of the Gear tooth spacing (GT) radially from the first line; usually this is all-time accomplished by radially copying the line from the first. 4.5 degrees is � of the gear tooth spacing (GT=eighteen degrees). 9) Now mirror a re-create of the involute curve around this second line, make certain you leave the original bend, thus copying the other side of the anfractuous ix degrees (i/2 GT) from the pitch circle (D) intersection with the involute. Drawing shows steps 7 - 9
10) Erase the radial lines, leaving the two anfractuous curves. Draw a line from the beginning of each anfractuous at the base circle to the center of the gear. Trim those lines to the Root Bore (DR) circle.
xi) Erase all the circles except the Root Diameter (DR) circle. Describe a curve from the outside tip of one involute to the other, which has a heart at 0,0 (the center of the gear) thus drawing the outside of the molar (the curve has the radius of RO). You now accept a completed gear tooth. 12) Radially re-create the completed gear tooth 19 times around the Root Diameter (DR) circumvolve, spacing the copies 18 degrees apart (GT), making 20 gear teeth (T) in full. xiii) Erase the Root Diameter (DR) circle and make a curve (or directly line) between ends of two teeth which has a eye at 0,0 (the heart of the gear). Purists volition annotation that I accept omitted the pocket-size fillet mostly drawn at the bottom of the root. I did not draw it because I will be milling this gear on my CNC milling motorcar and the endmill volition provide a fillet automatically. 14) Radially copy that bend or line around the gear as y'all did with the gear teeth. You now accept a completed involute gear.

An Application:

Milling a gear from flat stock with CNC If we want to mill this gear out of a flat plate on a CNC milling machine we need to effigy out what diameter endmill volition generate a minimum radius that won't interfere with the gear. If y'all are lucky plenty to have access to a laser or water abrasive jet machine and then you lot don't accept to worry about this. We tin can practice this graphically by drawing two meshing gears and either inserting a circle of the diameter of an endmill in the molar gullet - it should be apparent whether it interferes with the gear teeth (remember that we are concerned with the fillet the endmill produces, not the endmill itself, it tin overlap the other gear'southward tooth), or by inquiring in the cad program to the length of the root arc. In this case a i/sixteen" endmill will non interfere with the gear teeth meshing.

A rule of pollex that seems to work is to use an endmill with a diameter not larger than: DR * Pi / 2T = ane.1054 * PI / 40 = .0868"
A i/16" endmill is .0625" so information technology should piece of work, I have not tested this rule of thumb for all possible gears then revert to graphical assay if you accept any doubts. The drawing is imported into a CAM program and the g-code generated to manufactory the gear profile.

The gears milled with the method seem almost perfect and mesh perfectly in spite of the small steps that make upward the approximate involute.

Another Application, "Gauge Hobbing":

In his excellent commodity on "Spur Gears and Pinions" (HSM Apr 1999, Vol. 12, #2 pp. 8-15), John A. Cooper outlines a method for forming a gear with a cutting tool that is a round rack of the same pitch equally the gear. Part of his method entails cut the individual teeth, then lowering the cutter by half the circular pitch (CP) while keeping it engaged with the bare, thus rotating the bare while keeping the teeth in mesh and taking a second series of cuts, generating a good approximation of the involute.

Using what we take learned through cartoon the gear allows us to expand on the procedure and shows the relation between the rotation of the gear and the movement of the rack like cutter. On the lathe you make a cutter out of tool steel that is a circular rack of the same pitch as the gear, for the gear in the previous exercise the rack has 14.5 degree sides, the aforementioned equally an acme thread, so grind a tool bit the same for every bit for an acme thread. The grooves are pi/P (CP) apart, or 3.1415/sixteen (CP=.1963"), cutting flutes are milled and the cutter hardened. You then brand a gear blank of the desired size (aforementioned every bit the cartoon example, DO = ane.375") and mount information technology on a dividing caput, chuck the cutter you have made in the mill, and bring information technology down then the middle of the cutter is aligned with the midpoint on the gear. Take a cut(s) to the full tooth depth, across the width of the bare. Rotate the bare ane/8th of the gear tooth spacing (GT/8, xviii deg./8, 2.25 degrees), rather than leaving the gear in mesh with the cutter. Move the cutter in the direction of rotation by 1/8 x CP (1/8 * pi/P, .0245") Take another cut to total depth, repeat the process until you have made 8 passes. Retract the cutter against the management of rotation by pi/P (.1963") and begin the process over again until all the teeth are cut.

While this method is slow (unless you accept a CNC milling motorcar and 4th axis) if you practise eight passes, you tin can certainly go away with two or four passes and make a perfectly serviceable gear. Information technology does lend itself especially to making worm gears of almost perfect form.

Yous actually don't need to draw the gear for this method, just after drawing the gear you will have a better understanding of how the method works, and how far the blank needs to exist rotated and the cutter moved.

Concluding thoughts

Some other use of this method: Printed newspaper patterns (on label stock, particularly) could be used to grind unmarried point course tools for use in a fly cutter or on your shaper, for sawing wooden gears by manus with a jewelers saw or plasma cutting large gears from steel plate.

I'thousand sure the crafty reader will discover many other uses for this technique. This method can also be used with traditional drafting techniques, pencil and paper, but it will accept a much longer time. The original example of this method was taken from "Analysis and Design of Mechanisms" for drafting one tooth and copying each tooth equally you lot rotate a tracing around the circle.

I love manual drafting but there are so many inexpensive and free CAD programs available now that it would be a good time to upgrade if y'all are still using dividers and a t-square.

For those of y'all with a beloved of mathematics and figurer programming at that place is another way of generating the involute curve using polar coordinates, which lends itself to the generation of the curve in various programming languages or with spreadsheet and CAD macros. A quick search on the Internet using the term "Polar Involute" will render many pages dealing with that method.

If you are making meshing gears that take a large ratio (say a ten tooth gear and a 48 tooth gear) and yous draw them in mesh (separated between centers by half the pitch diameter of each gear), you will discover that the larger gear undercuts the teeth of the smaller gear, thus producing interference. There are strategies for dealing with this such equally increasing the centre altitude (thus backfire), stubbing the larger gear'south teeth, undercutting the smaller gear's teeth, etc, some farther research on your part will allow you to deal with this problem should it occur.

I promise this leaves you with a improve understanding of the geometry of an involute curve and a practical method of drawing gears for your projects.

• References: Analysis and Design of Mechanisms, Deane Lent, Prentice Hall, Inc.1961
• Machinery's Handbook, 27th ed., Industrial Printing, 2004
• "Spur Gears and Pinions", John A. Cooper, HSM April 1999, Vol. 12, #2

Copyright Nicholas Carter, 2007

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