Epicycloid and Epitrochoid Curves

A point on the pitch-circle of a gear that rolls around the outside of a stationary gear gives a curve called an Epicycloid.

If the point is not on the pitch-circle of the rolling gear, it has a generic name of Epitrochoid.

If the point is inside or outside the pitch-circle: Curtate and Prolate


Epitrochoid Curves are traced by a Point that is on a gear that rolls (orbits) around the outside of a fixed internal gear.

The shape of the curve depends on the ratio of the teeth. The ratio of the teeth is always a rational number. Therefore, the curve repeats, even if the gear ratio is a ratio of two prime numbers.

Epicycloid-Limacon

Equal Number of Gear Teeth.

This curve is called a Limaҫon.

If the Point is on the Pitch-Circle it is called a Cardioid.

The image shows four curves.

Epicycloid

One curve2s is given by a point on the pitch-circle of the orbiting gear - it is the Epicycloid

Epitrochoid

Two Protracted Epitrochoid curves3s4s are given by points outside the pitch-circle.

One Contracted Epitrochoid curves1s-red is given by a point inside the pitch-circle.

Epicycloid-Nephroid

The External Gear is half the diameter of the fixed gear.

This curve is called a Nephroid.

The image shows four curves.

Epicycloid

One curve is given by a point on the pitch-circle - it is the Epicycloid

Epitrochoid

Two curves are given by points outside the pitch-circle

One curve is given by a point inside the pitch-circle

The Gear ratio is 8: 1 There are four curves:  2 outside, one inside, one on the pitch-circle

The Gear ratio is 8: 1
There are four curves:
2 outside, one inside, one on the pitch-circle

Gear Ratio is 13:11 There is ONE curve. Use a Gearing FB to rotate the carrier 11 times to complete the Epicycloid Path.

Gear Ratio is 13:11
There is ONE curve.
Use a Gearing FB to rotate the carrier 11 times to complete the Epicycloid Path.

Gear Ratio 2:3 There is ONE Curve. Use a Gearing FB to rotate the carrier 3 times to complete the Epicycloid Path.

Gear Ratio 2:3
There is ONE Curve.
Use a Gearing FB to rotate the carrier 3 times to complete the Epicycloid Path.

Gear Ratio 24:6 There is ONE Curve. This is because the ratio reduces to 4:1

Gear Ratio 24:6
There is ONE Curve.
This is because the ratio reduces to 4:1

tog_minus        Epicycloid and the Hypocycloid Curves that give Straight-Lines

One of the most basic Epicycloid and Hypocycloid Curves that the Gear-Pairs can give is a perfect Straight Line.

Please see YouTube video to show designs.


Straight Line Curve with Hypocycloid Gears - Internal Mesh

The Video shows the Trace-Point follows a perfectly straight line.

You can see that the smaller gear has half the teeth of the other.

This is the standard Straight-Line Epicyclic Gear-Pair configuration.

Straight Line Curve with Epicycloid Gears - External Mesh - 1 - the Easy Way

There are two ways to assemble this mechanism in MechDesigner

The not easy, and the easy way.

This is the Easy Way

Straight Line Curve with Epicycloid Gears - External Mesh - 2 - the Not Easy Way

 

Spot the difference.

 

Tutorial and Reference Help Files for MechDesigner and MotionDesigner 13.2 + © Machine, Mechanism, Motion and Cam Design Software by PSMotion Ltd