Creating Gears and Splines in Wildfire 2.0

ProfilesMagazine.com

Creating Gears and Splines in Wildfire 2.0By Dan Marsalek, Marine Mechanical Corporation

Although many published methods exist for developing profiles of gear and spline teeth, the techniques are sometimes confusing and often inaccurate because they use only an approximation of the involute curve profile. The methods in this article clarify, expound, and improve on the current involute curve formulae commonly used. The methodology is based on Pro/ENGINEER Wildfire 2.0, but can also be easily adapted for use with WF1.0 and Pro/ENGINEER 2001. The equations for creating the involute datum curve are the same, although the extrusion and patterning of the final geometry are slightly different.

The Pro/ENGINEER user wishing to design a gear or spline tooth should start with the basics: the involute curve. An involute is described as the path of a point on a straight line, called the generatrix, as it rolls along a convex base curve (the evolute). The involute curve is most often used as the basis for the profile of a spline or gear tooth.

Here’s how to visualize the involute curve:

● Imagine a cylinder and a piece of string. ● Wrap the string tightly around the cylinder. ● Pull the string tight while unwinding it from the cylinder. ● Trace the end of the string as it is unwrapped. The result is an involute curve.

You should also acquaint yourselves with the standard features and terminology of gears and splines. The ANSI standards for gears (B6.1, B6.7) and splines (B92.1) are a good

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place to start, as is Machinery’s Handbook (1992, pp. 1787–2065). The AGMA standards are another source of good information for terms, symbols, equations, and definitions.

The following figure (taken from ANSI B92.1) illustrates a spline with standard dimensions and definitions.

Why should an involute profile be used in the design of a gear or spline tooth? Why not a straight edge? Some of the more important reasons:

● Conjugate action is independent of changes in center distance. Basically, if a driver gear with an involute tooth profile rotates at a uniform rate while acting on another gear with an involute tooth profile, the angular motion of the driven gear will be uniform. This is true even if the center-to-center distance is varied.

● The form of the basic rack tooth is straight-sided and therefore relatively simple. Thus, it can be accurately made. As a cutting tool, the rack tooth imparts high accuracy to the cut gear or spline tooth.

● One cutter can generate all gear or spline tooth numbers of the same pitch.

● The relative rate of motion between driven and driving gears having involute tooth curves is established by the diameters of their base circles.

● Contact between intermeshing involute teeth on a driving and driven gear is along a straight line that is tangent to the two base circles of these gears. This is called the “line of action.”

While several techniques can be used to create the involute tooth profile in Pro/ENGINEER, this article focuses on using datum curves by equation. The benefits of this method are that the involute curve profile is based on the exact geometric equations, it is



highly flexible in terms of the types of gears and curves that can be created, and it requires no additional Pro/ENGINEER modules (like ASX, AAX, BMX, etc.). In addition, the datum curve by equation technique allows you to use either Cartesian or cylindrical coordinate systems to create the involute curve profile. Finally, the curves generated by the methodology presented herein are automatically truncated at the major diameter, without the need for any additional operations (to trim the curve to size).

Why are there so many equations in this article? It’s great to know the final answer to a problem, but if you don’t know how you got to the solution, then you won’t be able to properly apply it. Additionally, the derivations of the equations validate the formulae that will be used in the relations editor to create the involute profile.

General Procedures for Involute Curve Creation

1. Set up parameters for key variables: a. Base diameter b. Pitch diameter c. Number of teeth d. Major (outer) tooth diameter e. Start angle (i.e., the angle from the horizontal axis in sketcher where the

involute starts) f. Circular tooth thickness or circular space width.

2. Create basic geometry in support of the spline or gear tooth. 3. Define the involute tooth profile with the datum curve by equation. 4. Create the tooth solid feature with a cut or protrusion.

a. Design vs. Manufacturing intent b. May need additional helical datum curves to sweep a helical gear teeth

5. Pattern the tooth around the centerline axis.

The trick is to know when to use Cartesian or cylindrical coordinates in creating the datum curve by equation.

● Use the cylindrical coordinate method if you want the easiest and most versatile method of involute creation, or if you have to use polar coordinates.

● Use Cartesian coordinates if you have to have the equations in terms of X, Y, and Z only.

Simply put, using the cylindrical coordinate system will be easier and quicker in most cases.

Deriving the Involute Datum Curve Equations — Cartesian Coordinates

The first step is to define terms and set up a sketch with the variables. The figure below presents a basic idea of what is involved in determining the equations for the involute datum curve.



Ri = Base dia./2

Ro = Major dia./2

Sα = arc length

SR = tangent line length at any point X,Y on the involute

SRo = tangent line length at major diameter on involute

β = angle from start of involute to tangent point on base circle

Xc,Yc = tangent point on base circle corresponding to tangent line SR

Start_angle = angle from the horizontal axis to the start of the involute curve

For simplicity, we will assume a start angle of 0° and remove it from the formulae.

From basic trigonometry:



From the geometry, the equation for XR and YR can be derived:

From the Pythagorean Theorem:

Substitute and simplify the equations to get XR and YR in terms of Ri and β. Start by

substituting for Xc and SR in the original equation for XR:

Substitute for Yc and SR in the original equation for YR:

Take the equations for XR and YR, above, and plug them into the equation for RO:



Expand the squared terms:

Consolidate terms that have Ri2, and Ri

2 * β:

Remember a basic trigonometric identity formula:

Substitute for the identity, and combine like terms (which add to 0). The equation for Ro

becomes:



Combining terms that have Ri2:

Squaring Ro gives us:

Squaring the square root term:

Rearranging the equation to isolate β:



We need to define a term, α, in terms of Ri and Ro, so that we can solve the parametric

equation for the creation of the datum curve. We also need to evaluate β over its full range (from Ri to Ro) to derive the involute curve, so we multiply by “t” in the equation

(t varies linearly from 0 to 1):

Substituting for β:

But since we want radians, we multiply by 2* π /360:



We need the parametric equations for X and Y (and Z) in terms of Ri and α. We will use

XR and YR as the basis, substituting α for β (and multiplying the α terms by 360/2*π

because we need degrees here):

Deriving the parametric equation for X:

Deriving the parametric equation for Y:

Finally, Z = 0 (since we wish to create a 2-D planar curve)



So, the relation equations used in creating the involute profile datum curve will be:

solve

for α

Z = 0

Remember that all variables (α, Ro, Ri) must be predefined. Since we don’t know α yet,

just preset it to a value of 1 using parameters, or set it to 1 before the involute curve equations in the relation equation editor.

If we include a start angle of some value other than 0°, the equations become:

solve

for α



Z = 0

Deriving the Involute Datum Curve Equations — Cylindrical Coordinates

Ri = Base dia./2

Ro = Major dia./2

R = Radius to any point on the involute curve

Sα = arc length from start of the involute to the tangent point

SR = tangent line length at any point X,Y on the involute

SRo = tangent line length at major diameter on involute

β = angle from start of involute to tangent point on base circle

θ = angle from start of involute to any point on the involute between Ro and Ri

α = angle from a point on the involute to the tangent point on base circle

Start_angle = angle from the horizontal axis to the beginning of the involute curve

The geometry associated with the involute curve in a cylindrical coordinate system is shown below.



Again, we are setting the start angle to 0° for simplicity.

The following equations are based on the geometry of the involute setup:

The relationship between SR and SRo varies linearly, so if we denote a variable t that

goes from 0 to 1 as R goes from 0 to Ro, then:



It is important to note that SR = SRo when R = Ro (because t=1).

By the Pythagorean Theorem:

By observation and the Pythagorean Theorem:

Substituting (SRo * t) for SR into the equation defining R:

We need to find β in terms of Ri and Ro. Start with the equation for SR found previously:

Isolate the β term and substitute (SRo * t) for SR:



Solving the equations to get α:

Substitute (SRo * t) for SR:

Substituting for β and α in the equation for θ:

As in the case for the equations for Cartesian involute curves, we still want the curve to be 2-D and planar, so:

Z = 0

We need to make the equations parametric based on Ro and Ri and t (which varies

linearly from 0 to 1), so we create a variable γ, similar to the α term in the Cartesian Coordinate equations, but based on SR instead of β:



Substituting for SRo:

Substituting γ for SRo into the equations for R gives us:

A similar substitution for SRo in the equation for θ:

So, the relation equations used in the creation of the involute profile datum curve will be:



Solve

for γ

Z = 0

Note: to account for a start angle ≠ 0, use:

As with the Cartesian coordinate method, remember to predefine γ, Ro, and Ri before

solving the relations.

Using the Cylindrical Coordinate System to Create an Involute Curve

1. Create the part parameters (major_dia, pitch_dia, minor_dia, base_dia, tooth_thick (or space_width), no_of_teeth, start_angle, gamma).



(Click to enlarge)

2. Create the base cylinder geometry. Use the major_dia parameter as the OD of the cylinder.



(Click to enlarge)



(Click to enlarge)

3. Make sure you have a coordinate system already established. It should be located on the centerline axis at one end of the cylinder you have created. (It helps if the coordinate system has the Z direction along the centerline axis.) Create the involute datum curve by using Datum Curve, By Equation. Choose a cylindrical coordinate system.



(Click to enlarge)

4. Create a datum curve (by sketch) that represents the entire cut for the tooth profile. Create a centerline, and then mirror the involute curve to make the two sides of the cut. Set the two curves apart by using a construction arc at the pitch diameter to represent the circular space width. Dimension the arc as a perimeter by selecting the angular dimension and choosing Convert, To Perimeter from the Edit dropdown menu. Make the top and bottom sides of the cut using an arc and selecting the end points to be symmetric about the centerline.

(Click to enlarge)

5. OK the section. Then extrude a cut axially along the cylinder based on the profile of the datum curve in Step 4.



(Click to enlarge)

6. Pattern the cut axially around the cylinder. Voila! Your gear/spline is now ready for use!



(Click to enlarge)

For Further Reference

1. The Society of Automotive Engineers, ANSI Standard B92.1-1996, 1996.

2. www.cadquest.com/books/pdf/gears.pdf, Involute Gear Design Tutorial.

3. PTC Knowledgebase, Suggested Techniques for:

● Creation of an Involute Gear Cutting (3 Methods) ● Creating a Cylindrical Gear with Helical Teeth ● Creating an Involute Curve

4. Machinery’s Handbook, 24th ed., 1992, pp. 1787-2065.

5. Shigley and Mishke, Mechanical Engineering Design, 5th ed., 1989, pp. 527-584

6. Roy Beardmore, www.roymech.co.uk/Useful_Tables/Drive/Gears.html, 2004.

Dan Marsalek is a design engineer with Marine Mechanical Corp. in Cleveland, Ohio, specializing in 3-D modeling and nonlinear FEA of complex mechanisms for naval vessels. Dan is currently vice president of the Ohio Pro/E User Group (NOPUG). This article is based in part on his


http://www.roymech.co.uk/Useful_Tables/Drive/Gears.html


presentation at the 2005 PTC/USER World Event.


Creating Gears and Splines in Wildfire 2.0

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