Variable Stroke Pt. 1
I’ve been experimenting with variable stroke linkages for a while, with most of my new prototypes being 3d printed. These linkages have two inputs – the crankshaft and the stroke amplitude input – so I’ve been using gears to drive the two from one input.
Traditional gear manufacture involves using off-the-shelf tools to cut gears from blanks, and proper use of the tools will give accurate geometry. But 3d printing the gears requires an exact model of the involute tooth surface, which is a bit more tricky. Solidworks includes a “gear” template, but this is just a placeholder, not an accurate model. Some models can be found online, but these are almost always in imperial units. I wanted a parametric model of a precise involute cylindrical gear, using the metric standard, so I made my own.
There are 5 key numbers that define a given gear:
- – module
- – number of teeth
- – pressure angle
- – helix angle
- – profile shift
I’ll discuss the last two in a later post. For now we can assume they are equal to zero, so that the model is only dependent on the first three.
A metric gear has two primary tooth size measurements aside from : – the addendum height, and – the root clearance. These are shown below in a sketch of an “unrolled” tooth. Typically, and . Four important dimensions associated with the gear are:
- – the pitch diameter
- – the tip diameter
- – the root diameter
- – the base circle diameter
Involute gears use a tooth profile following an involute curve – this is by definition the curve obtained by unrolling a string from a circle. The “string” is always tangent to it’s “base circle” and perpendicular to the curve obtained, as shown below for a base circle of radius . The position of point is . Point on the involute is then:
This equation can be used directly in Solidworks as an equation curve defining the tooth face, however it only defines one side of one tooth. Ideally the modelled tooth will be symmetric about the x-axis, so the involute needs to be offset (rotated around the base circle) by the appropriate amount. We also need to know when to start and end the curve.
The rotated involute curve can be formulated by multiplying the involute equation above by a rotation matrix , which rotates the curve counter-clockwise about the origin by radians. The generic equation for an involute of a base circle with radius , rotated by angle is:
The involute curve needs to be rotated such that the tooth axis lies on the x-axis, as depicted below. This angle can be separated into two components:
- The half-tooth-width angle
- The involute angle
So , and the involute should be rotated clockwise by this angle, so . The equation for this curve is:
For the opposite of the tooth, the involute “string” is beign unrolled clockwise: simply take . The curve is then rotated counter-clockwise, so take .
What are the limits of the curve? From the above diagram, the distance between the origin and is . Then . But we also have , so the angle where point is at a diameter is given by:
The maximum point of the curve (at the tip) is then given by .
The minimum point (at the root) is given by .
The tooth profile is then formed by tracing the involute curve between these limits. Except for some gears ( – see below) the root is actually below the base circle – and the involute is not defined below the base circle.
In this case I model the tooth flank as a straight, radial line. In reality the tooth flank is sometimes undercut, and I will go into this in another post.