__Cycloidal____ Gear__

Watches
and clocks traditionally use **cycloidal** gearing.
This section will focus on generation of the cycloidal
gear profile. More details on the design can be found in British Standard 978
Part 2: *Cycloidal** Type Gears..*

Development
of the mathematical equations for the cycloidal curve
uses simple

rotation.
The equations can be obtained referring to the following figure. The epicycloid starts from the pitch circle on which the
generating circle rolls.

**Figure
4.4.3.2.1 **Development of an epicycloidal
curve

Considering
two points, P and Q, on the generating circle, while the point Q moves to the postion Q’, the point P moves to the position P’. Since the
arc lengths of PQ and P’Q’ are same, the position P’ can be determined by
rotating the position Q’.

(4.4.3.1)

where r
and r_{g} are radii of the pitch circle and
generating circle, respectively. Note that with negative value of r_{g} the curve becomes hypocycloid.

The
gear profile below pitch circle can be straight radial lines or can be
generated by the hypocycloid.

Let’s
take a case with N = 26 and DP = 4 as in the case of involute
gear.

1.
Sketch Curve >
create the outside circle

2.
Repeat this for
the pitch, base and root circles

3.
Change the
dimension name for outside circle to OD, the first offset dimension to
ADDENDUM, the second offset dimension to DEDENDUM, and the last offset
dimension to CLEARANCE

4.
**Tools** > **Para****meter** > **Add** (**+**) > add DG
(diameter of generating circle) and its value 2 > OK

5.
**Tools** > **Program** > in the RELATIONS section,
enter the relations as below > save and exit

**RELATIONS**

PD =
OD - 2*ADDENDUM

BD =
PD - 2*DEDUNDUM

RD =
BD - 2*CLEARANCE

**END RELATIONS**

6.
**Datum Curve**
> **From Equations** / Done > pick
the default coordinates > **Cartesian**
> enter the equation as below > File > Save > Exit > OK

7.
Let’s create the
hypocycloid below the pitch circle using the datum curve with same equation as
below. The curves appear as

8.
Edit > Feature
Operations > Copy > Move / Done > select both curves > OK (select)
> **Rotate** > Crv/Edge/Axis
> pick the axis at the center > select the direction and ** **(i.e., b in Figure
4.4.3.2.8) > CR > Done Rotate > OK (feature)

9.
Pick both curves
in Model Tree > **Mirror** > pick
the horizontal datum plane > OK (feature). Now, the curves appear as

10. **Extrude** > create a circular plate with **UseEdge** and outside diameter,
and then the thickness of 1.0

11. Extrude > using **UseEdge** again, select the curves
for boundary of the cut > use **Trim to
Entity** to trim the corners (**Dynamic Trim** may leave stranded curves giving error messages at
exit) > **To Next** for depth > **Remove Materials** > selection the
directions of cut properly > Done (feature). One
cut appears

12. Pick the cut (if not highlighted) > **Pattern** > select **Axis** for type in dropdown list >
enter 26 for the number of patterns and the angle 360/26 > Done (feature). The final cycloidal
gear appears as

**Figure
4.4.3.2.2 **Cycloidal gear and its profile

13. Complete the relations and inputs in the program as
below > save and exit > Yes (to incorporate the changes) > Enter >
select all values and then enter 26, 4, and 2.

14. **Regenerate** > try to play with these input
values to change the gear shapes.

The
following shows the profiles for various diameters of generating circle. Note
that as DG increases, the profile below pitch circle becomes almost straight
just like real teeth that is often approximated by
straight radial lines.

**Figure
4.4.3.2.3 **Cycloidal gear profiles for various diameter of generating
circle